LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORYcomplex/Files/slidesptempesta.pdf · LECTURES ON...
Transcript of LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORYcomplex/Files/slidesptempesta.pdf · LECTURES ON...
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
LECTURES ONGROUPS, ENTROPIES AND NUMBER
THEORY
Piergiulio Tempesta
Universidad Complutense de Madridand
Instituto de Ciencias Matematicas (ICMAT),Madrid, Spain.
TEMPLETON SCHOOL ON FOUNDATIONS OF COMPLEXITY
Rio de Janeiro, October 14 - 15, 2015
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
LECTURE I
On the Mathematical Foundations of theTheory of Generalized Entropies
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
OutlineIntroduction: A Foundational Perspective
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Formal Group TheoryDefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Strong and weak composabilityStrong and weak composabilityGeneral properties of the trace-form class
Groups, Difference Operators and EntropiesDelta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operatorsPiergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
SK and
Composability
axioms Generalized Entropies
Group-theoretical
Structure
Delta Operators Formal group
theory
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
A Foundational Perspective
Statement of the problem
I What are the mathematical foundations of the notion ofentropy?
I The main idea is to propose a group-theoretical approach to thetheory of generalized entropies
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
A Foundational Perspective
Statement of the problem
I What are the mathematical foundations of the notion ofentropy?
I The main idea is to propose a group-theoretical approach to thetheory of generalized entropies
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
A Foundational Perspective
Statement of the problem
I What are the mathematical foundations of the notion ofentropy?
I The main idea is to propose a group-theoretical approach to thetheory of generalized entropies
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Different foundational approaches
This problem has been addressed from many different perspectives:
I Large deviation theory
I Maximum entropy principle
I Superstatistics
I Information Theory:
Divergences (Kullback-Leibler, Renyi, Bregman, Csiczar, α-,β-, etc.)
Exponential families
Information Geometry (Riemannian manifolds, conformal geometry),
etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Different foundational approaches
This problem has been addressed from many different perspectives:
I Large deviation theory
I Maximum entropy principle
I Superstatistics
I Information Theory:
Divergences (Kullback-Leibler, Renyi, Bregman, Csiczar, α-,β-, etc.)
Exponential families
Information Geometry (Riemannian manifolds, conformal geometry),
etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Different foundational approaches
This problem has been addressed from many different perspectives:
I Large deviation theory
I Maximum entropy principle
I Superstatistics
I Information Theory:
Divergences (Kullback-Leibler, Renyi, Bregman, Csiczar, α-,β-, etc.)
Exponential families
Information Geometry (Riemannian manifolds, conformal geometry),
etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Different foundational approaches
This problem has been addressed from many different perspectives:
I Large deviation theory
I Maximum entropy principle
I Superstatistics
I Information Theory:
Divergences (Kullback-Leibler, Renyi, Bregman, Csiczar, α-,β-, etc.)
Exponential families
Information Geometry (Riemannian manifolds, conformal geometry),
etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Characterization theorems
Let W ∈ N/0. Let piWi=1, with pi ≥ 0 for all i = 1, . . . ,W ,∑Wi=1 pi = 1 denote a discrete probability distribution, where PW
represents the set of all discrete probability distributions with W entries.The Boltzmann-Gibbs entropy
SBG : PW −→ R+ ∪ 0
SBG [p] :=W∑i=1
pi ln1
p1, piWi=1 ∈ PW
has been characterized by several existence and uniqueness theorems,proved in the last century.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Khinchin’s theorem
(A. I. Khinchin, Mathematical foundations of information theory, 1957).Given two systems A and B, let pij(A ∪ B) denote its joint probabilitydistribution, i = 1, . . . ,WA, j = 1, . . . ,WB . Also,
pi (A) :=
WB∑j=1
pij(A ∪ B).
Let S(B | A) denote the conditional entropy associated with theconditional probability distribution pij(B | A) = pij(A ∪ B)/pi (A).Theorem 1.1.i) Let S be a continuous function of all its arguments (p1, . . . , pW ).i) Assume that it takes its maximum for the equiprobability distribution,i.e. when pi = 1/W , for i = 1, . . . ,W .iii) Also, assume that S(p1, . . . , pw , 0) = S(p1, . . . , pw ).iv) Given two systems A and B, S(A ∪ B) = S(A) + S(B | A).Then the unique entropy satisfying (i)-(iv) is Boltzmann’s entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Abe’s theorem
S. Abe, Phys. Lett. A, 271, 74-79 (2000).Theorem 1.2.i) Let S be a continuous function of all its arguments (p1, . . . , pW ).i) Assume that it takes its maximum for the equiprobability distribution,pi = 1/W , for i = 1, . . . ,W .iii) Also, assume that S(p1, . . . , pw , 0) = S(p1, . . . , pw ).iv) Given two systems A and B,S(A ∪ B) = S(A) + S(B | A) + (1− q)S(A)S(B | A).Then the unique entropy satisfying the axioms (i)-(iv) is Tsallis’entropy:
Sq[p] :=
∑Wi=1 pq
i − 1
1− q, q > 0.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
dos Santos’ theorem
R. J. V. dos Santos, J. M. Phys. 38, 4104 (1997)This theorem generalizes the original Shannon’s uniqueness theorem forthe Boltzmann entropy.
Theorem 1.3i) Let S be a continuous function of all its arguments (p1, . . . , pW ).ii) For the equiprobability distribution, pi = 1/W , for i = 1, . . . ,W , S isa monotonic increasing function of W , i.e.
S =W 1−q − 1
1− q.
iii) Given two statistically independent systems A and B,S(A ∪ B) = S(A) + S(B) + (1− q)S(A)S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
iv) If W = WL + WM , with
PL =
WL∑i=1
pi , (WL terms)
PM =W∑
i=WL+1
pi , (WM terms)
(hence PL + PM = 1)we have:
S [p] = S(pL, pM) + (pL)qS
(pi
pL
)+ (pM)qS
(pi
pM
).
Then, the unique entropy satisfying the axioms (i)-(iv) is Tsallis’entropy Sq.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
The modern Shannon-Khinchin formulation
I (SK1) (Continuity). The function S(p1, . . . , pW ) is continuous withrespect to all its arguments
I (SK2) (Maximum principle). The function S(p1, . . . , pW ) takes itsmaximum value over the uniform distribution pi = 1/W ,i = 1, . . . ,W .
I (SK3) (Expansibility). Adding an impossible event to a probabilitydistribution does not change its entropy:S(p1, . . . , pW , 0) = S(p1, . . . , pW ).
I (SK4) (Additivity). Given two subsystems A, B of a statisticalsystem,
S(A ∪ B) = S(A) + S(B | A).
In particular, if they are statistically independent,
S(A ∪ B) = S(A) + S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
The modern Shannon-Khinchin formulation
I (SK1) (Continuity). The function S(p1, . . . , pW ) is continuous withrespect to all its arguments
I (SK2) (Maximum principle). The function S(p1, . . . , pW ) takes itsmaximum value over the uniform distribution pi = 1/W ,i = 1, . . . ,W .
I (SK3) (Expansibility). Adding an impossible event to a probabilitydistribution does not change its entropy:S(p1, . . . , pW , 0) = S(p1, . . . , pW ).
I (SK4) (Additivity). Given two subsystems A, B of a statisticalsystem,
S(A ∪ B) = S(A) + S(B | A).
In particular, if they are statistically independent,
S(A ∪ B) = S(A) + S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
The modern Shannon-Khinchin formulation
I (SK1) (Continuity). The function S(p1, . . . , pW ) is continuous withrespect to all its arguments
I (SK2) (Maximum principle). The function S(p1, . . . , pW ) takes itsmaximum value over the uniform distribution pi = 1/W ,i = 1, . . . ,W .
I (SK3) (Expansibility). Adding an impossible event to a probabilitydistribution does not change its entropy:S(p1, . . . , pW , 0) = S(p1, . . . , pW ).
I (SK4) (Additivity). Given two subsystems A, B of a statisticalsystem,
S(A ∪ B) = S(A) + S(B | A).
In particular, if they are statistically independent,
S(A ∪ B) = S(A) + S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
The modern Shannon-Khinchin formulation
I (SK1) (Continuity). The function S(p1, . . . , pW ) is continuous withrespect to all its arguments
I (SK2) (Maximum principle). The function S(p1, . . . , pW ) takes itsmaximum value over the uniform distribution pi = 1/W ,i = 1, . . . ,W .
I (SK3) (Expansibility). Adding an impossible event to a probabilitydistribution does not change its entropy:S(p1, . . . , pW , 0) = S(p1, . . . , pW ).
I (SK4) (Additivity). Given two subsystems A, B of a statisticalsystem,
S(A ∪ B) = S(A) + S(B | A).
In particular, if they are statistically independent,
S(A ∪ B) = S(A) + S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
A possible unifying principle: The group-theoreticalapproach
We wish to find, possibly, a unifying principle, that would be responsibleof the many properties of a generalized entropy:
Thermodynamics: Legendre structure, Extensivity, SK axioms, etc.
Classical and Quantum Information Theory: Generalized divergences,etc.This principle should allow to classify all the known entropies and allowto design new ones.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Beyond the SK formulation: the Composability Axiom
P. T., Beyond the SK formulation: the Composability axiom and theUniversal-group entropy, arXiv: 1407.3807 Annals of Physics (in press)
I The notion of entropy should be properly defined just in terms ofmacroscopic configurations of a system (without the need for amicroscopic description of the associated dynamics).
I Foundational property: an entropy should be a coarse-grainedquantity (Gell-Mann); if not so, the concept of entropy would besimply empty.
I Even the second law of thermodynamics loses any meaning if notreferred to the evolution of macroscopic subsystems: the entropywould stay invariant if defined on microscopic configurations.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Beyond the SK formulation: the Composability Axiom
P. T., Beyond the SK formulation: the Composability axiom and theUniversal-group entropy, arXiv: 1407.3807 Annals of Physics (in press)
I The notion of entropy should be properly defined just in terms ofmacroscopic configurations of a system (without the need for amicroscopic description of the associated dynamics).
I Foundational property: an entropy should be a coarse-grainedquantity (Gell-Mann); if not so, the concept of entropy would besimply empty.
I Even the second law of thermodynamics loses any meaning if notreferred to the evolution of macroscopic subsystems: the entropywould stay invariant if defined on microscopic configurations.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Beyond the SK formulation: the Composability Axiom
P. T., Beyond the SK formulation: the Composability axiom and theUniversal-group entropy, arXiv: 1407.3807 Annals of Physics (in press)
I The notion of entropy should be properly defined just in terms ofmacroscopic configurations of a system (without the need for amicroscopic description of the associated dynamics).
I Foundational property: an entropy should be a coarse-grainedquantity (Gell-Mann); if not so, the concept of entropy would besimply empty.
I Even the second law of thermodynamics loses any meaning if notreferred to the evolution of macroscopic subsystems: the entropywould stay invariant if defined on microscopic configurations.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
I Therefore, when composing two different systems, defined overany probability distribution piWi=1, we should have:
S(A ∪ B) = Φ(S(A),S(B); η) (1)
where η is a possible set of parameters.
I In additionS(A ∪ B) = S(B ∪ A).
I Another crucial property:
S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )
I Finally, when we compose a system A with another system B, suchthat S(B) = 0, we should have that
S(A ∪ B) = S(A).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
I Therefore, when composing two different systems, defined overany probability distribution piWi=1, we should have:
S(A ∪ B) = Φ(S(A),S(B); η) (1)
where η is a possible set of parameters.
I In additionS(A ∪ B) = S(B ∪ A).
I Another crucial property:
S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )
I Finally, when we compose a system A with another system B, suchthat S(B) = 0, we should have that
S(A ∪ B) = S(A).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
I Therefore, when composing two different systems, defined overany probability distribution piWi=1, we should have:
S(A ∪ B) = Φ(S(A),S(B); η) (1)
where η is a possible set of parameters.
I In additionS(A ∪ B) = S(B ∪ A).
I Another crucial property:
S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )
I Finally, when we compose a system A with another system B, suchthat S(B) = 0, we should have that
S(A ∪ B) = S(A).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
I Therefore, when composing two different systems, defined overany probability distribution piWi=1, we should have:
S(A ∪ B) = Φ(S(A),S(B); η) (1)
where η is a possible set of parameters.
I In additionS(A ∪ B) = S(B ∪ A).
I Another crucial property:
S(A ∪ (B ∪ C )) = S((A ∪ B) ∪ C )
I Finally, when we compose a system A with another system B, suchthat S(B) = 0, we should have that
S(A ∪ B) = S(A).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
The Shannon-Khinchin axioms (SK1)-(SK3) are basic properties, nonnegotiable.
However, they are not sufficient for thermodynamical purposes.
We need a further requirement, that replaces the axiom (SK4) in fullgenerality: the Composability Axiom.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
The Composability Axiom
Definition 1.4. An entropy S is (strictly) composable if there exists asmooth function of two real variables Φ(x , y) such that (C1)
S(A ∪ B) = Φ(S(A),S(B); η) (2)
where A ⊂ X and B ⊂ X are two statistically independent systems,defined over any probability distribution piWi=1, with the furtherproperties(C2) Symmetry:
Φ(x , y) = Φ(y , x). (3)
(C3) Associativity:
Φ(x ,Φ(y , z)) = Φ(Φ(x , y), z). (4)
(C4) Null-composability:
Φ(x , 0) = x . (5)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
Composition of two arbitrary systems
Remark. Condition (2) can be formulated in more general terms.Consider the case of a composite system A ∪ B arising from two systemsnot statistically independent, with a conditional probability distributionpij(B | A) := pij(A ∪ B)/pi (A).Here pij(A ∪ B), i = 1, . . . ,WA, j = 1, . . . ,WB denotes the jointprobability distribution for the composite system A ∪ B, and pi (A) is
the marginal probability distribution pi (A) =∑WB
j=1 pij(A ∪ B).In this general context we postulate the relation
S(A ∪ B) = Φ(S(A),S(B | A); η), (6)
where S(B | A) denotes the conditional entropy associated with theconditional distribution pij(B | A). Equation (6) reduces to the relation(2) in the case of statistically independent subsystems. The relation (6)generalizes the original axiom (SK4) for the case of systems notindependent.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Different foundational approachesThe uniqueness theorems for the Boltzmann-Gibbs entropyThe SK approachBeyond the SK formulation: the Composability Axiom
How to realize the composability axiom?
According to the Composability Axiom, we are interested to the generalclass of functions
Φ : R2 −→ R
such that the properties (C 1)-(C 4) apply. Surprisingly enough, there is afull mathematical theory that deals with such functions: the FormalGroup Theory. It was developed in the context of algebraic topologystarting from the seminal work of Bochner in 1946, and is still an activeresearch area.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Formal Group Theory: a crash course
The theory of formal groups offers a natural language for formulatingthe theory of generalized entropies.S. Bochner, Formal Lie groups, Annals of Mathematics 47, 192–201(1946).M. Hazewinkel, Formal Groups and Applications, Academic Press, NewYork, 1978.Many important mathematicians have been working in this area: S. P.Novikov, D. Quillen, J. P. Serre, G. Faltings, etc.Nowadays, it has a prominent role in fields as
I Algebraic topology,
I The theory of elliptic curves
I Arithmetic number theory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Formal Group Theory: a crash course
The theory of formal groups offers a natural language for formulatingthe theory of generalized entropies.S. Bochner, Formal Lie groups, Annals of Mathematics 47, 192–201(1946).M. Hazewinkel, Formal Groups and Applications, Academic Press, NewYork, 1978.Many important mathematicians have been working in this area: S. P.Novikov, D. Quillen, J. P. Serre, G. Faltings, etc.Nowadays, it has a prominent role in fields as
I Algebraic topology,
I The theory of elliptic curves
I Arithmetic number theory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Formal Group Theory: a crash course
The theory of formal groups offers a natural language for formulatingthe theory of generalized entropies.S. Bochner, Formal Lie groups, Annals of Mathematics 47, 192–201(1946).M. Hazewinkel, Formal Groups and Applications, Academic Press, NewYork, 1978.Many important mathematicians have been working in this area: S. P.Novikov, D. Quillen, J. P. Serre, G. Faltings, etc.Nowadays, it has a prominent role in fields as
I Algebraic topology,
I The theory of elliptic curves
I Arithmetic number theory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Definitions
I Definition 1.5. Let R be a commutative ring with identity, andR x1, x2, .. be the ring of formal power series in the variables x1,x2, ... with coefficients in R.
I A commutative one-dimensional formal group law over R is aformal power series in two variables Φ (x , y) ∈ R x , y of the formΦ (x , y) = x + y + terms of higher degree, such that
i) Φ (x , 0) = Φ (0, x) = x
ii) Φ (Φ (x , y) , z) = Φ (x ,Φ (y , z)).
I When Φ (x , y) = Φ (y , x), the formal group law is said to becommutative.
I Proposition 1.6. Form any formal group law Φ(x , t), there existsan inverse formal series ϕ (x) ∈ R x such that Φ (x , ϕ (x)) = 0.
I This is the reason why we talk about formal group laws.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Definitions
I Definition 1.5. Let R be a commutative ring with identity, andR x1, x2, .. be the ring of formal power series in the variables x1,x2, ... with coefficients in R.
I A commutative one-dimensional formal group law over R is aformal power series in two variables Φ (x , y) ∈ R x , y of the formΦ (x , y) = x + y + terms of higher degree, such that
i) Φ (x , 0) = Φ (0, x) = x
ii) Φ (Φ (x , y) , z) = Φ (x ,Φ (y , z)).
I When Φ (x , y) = Φ (y , x), the formal group law is said to becommutative.
I Proposition 1.6. Form any formal group law Φ(x , t), there existsan inverse formal series ϕ (x) ∈ R x such that Φ (x , ϕ (x)) = 0.
I This is the reason why we talk about formal group laws.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Definitions
I Definition 1.5. Let R be a commutative ring with identity, andR x1, x2, .. be the ring of formal power series in the variables x1,x2, ... with coefficients in R.
I A commutative one-dimensional formal group law over R is aformal power series in two variables Φ (x , y) ∈ R x , y of the formΦ (x , y) = x + y + terms of higher degree, such that
i) Φ (x , 0) = Φ (0, x) = x
ii) Φ (Φ (x , y) , z) = Φ (x ,Φ (y , z)).
I When Φ (x , y) = Φ (y , x), the formal group law is said to becommutative.
I Proposition 1.6. Form any formal group law Φ(x , t), there existsan inverse formal series ϕ (x) ∈ R x such that Φ (x , ϕ (x)) = 0.
I This is the reason why we talk about formal group laws.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Definitions
I Definition 1.5. Let R be a commutative ring with identity, andR x1, x2, .. be the ring of formal power series in the variables x1,x2, ... with coefficients in R.
I A commutative one-dimensional formal group law over R is aformal power series in two variables Φ (x , y) ∈ R x , y of the formΦ (x , y) = x + y + terms of higher degree, such that
i) Φ (x , 0) = Φ (0, x) = x
ii) Φ (Φ (x , y) , z) = Φ (x ,Φ (y , z)).
I When Φ (x , y) = Φ (y , x), the formal group law is said to becommutative.
I Proposition 1.6. Form any formal group law Φ(x , t), there existsan inverse formal series ϕ (x) ∈ R x such that Φ (x , ϕ (x)) = 0.
I This is the reason why we talk about formal group laws.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Definitions
I Definition 1.5. Let R be a commutative ring with identity, andR x1, x2, .. be the ring of formal power series in the variables x1,x2, ... with coefficients in R.
I A commutative one-dimensional formal group law over R is aformal power series in two variables Φ (x , y) ∈ R x , y of the formΦ (x , y) = x + y + terms of higher degree, such that
i) Φ (x , 0) = Φ (0, x) = x
ii) Φ (Φ (x , y) , z) = Φ (x ,Φ (y , z)).
I When Φ (x , y) = Φ (y , x), the formal group law is said to becommutative.
I Proposition 1.6. Form any formal group law Φ(x , t), there existsan inverse formal series ϕ (x) ∈ R x such that Φ (x , ϕ (x)) = 0.
I This is the reason why we talk about formal group laws.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
The previous definition can be naturally extended to the case ofn-dimensional formal group laws.
Definition 1.7.Let R be a commutative ring with identity, and R x1, x2, .. be the ringof formal power series in the variables x1, x2, ... with coefficients in R.An n-dimensional formal group law over R is an n-tuple
Φ = (Φ1, . . . ,Φn)
of power series Φi (x,y), i = 1, . . . , n, where x= (x1, . . . , xn), withcoefficients in R, such that
Φ(x, 0) = x
Φ(x,Φ(y, z)) = Φ(Φ(x, y), z)
If Φ(x, y) = Φ(y, x), the n- dimensional formal group law is said to becommutative.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Formal groups and formal group laws
I The relevance of formal group laws relies first of all on their closeconnection with classical group theory.
I A formal group law Ψ(x , y) defines a functor F : AlgR −→ Group,where AlgR denotes the category of commutative unitary algebrasover R and Group denotes the category of groups. The functor F isby definition the formal group (sometimes called the formal groupscheme) associated to the formal group law Ψ.
I We can construct a formal group law of dimension n from anyalgebraic group or Lie group of the same dimension n, by takingcoordinates at the identity and writing down the formal power seriesexpansion of the product map. An important special case of this isthe formal group law of an elliptic curve (or abelian variety).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Formal groups and formal group laws
I The relevance of formal group laws relies first of all on their closeconnection with classical group theory.
I A formal group law Ψ(x , y) defines a functor F : AlgR −→ Group,where AlgR denotes the category of commutative unitary algebrasover R and Group denotes the category of groups. The functor F isby definition the formal group (sometimes called the formal groupscheme) associated to the formal group law Ψ.
I We can construct a formal group law of dimension n from anyalgebraic group or Lie group of the same dimension n, by takingcoordinates at the identity and writing down the formal power seriesexpansion of the product map. An important special case of this isthe formal group law of an elliptic curve (or abelian variety).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Formal groups and formal group laws
I The relevance of formal group laws relies first of all on their closeconnection with classical group theory.
I A formal group law Ψ(x , y) defines a functor F : AlgR −→ Group,where AlgR denotes the category of commutative unitary algebrasover R and Group denotes the category of groups. The functor F isby definition the formal group (sometimes called the formal groupscheme) associated to the formal group law Ψ.
I We can construct a formal group law of dimension n from anyalgebraic group or Lie group of the same dimension n, by takingcoordinates at the identity and writing down the formal power seriesexpansion of the product map. An important special case of this isthe formal group law of an elliptic curve (or abelian variety).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
I Viceversa, given a formal group we can construct a Lie algebra.
I Construction. Let us write
Φ(x, y) = x + y + Φ2(x, y) + Φ3(x, y) + . . .
where Φ2(x , y) denotes the quadratic part of the formal group law,Φ3(x , y) the cubic part, and so on.Any n-dimensional formal group law induces an n-dimensional Liealgebra over the same ring R, by means of the formula
[x, y] := Φ2(x, y)−Φ2(y, x).
I Lie Groups −→ Formal Group Laws ←→ Lie Algebras
I This correspondence is true over fields F of characteristic zero, andis lost if char(F) 6= 0. (See J.-P. Serre, Lie Algebras and Lie Groups,LNM 1500, Springer, 1992).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
I Viceversa, given a formal group we can construct a Lie algebra.
I Construction. Let us write
Φ(x, y) = x + y + Φ2(x, y) + Φ3(x, y) + . . .
where Φ2(x , y) denotes the quadratic part of the formal group law,Φ3(x , y) the cubic part, and so on.Any n-dimensional formal group law induces an n-dimensional Liealgebra over the same ring R, by means of the formula
[x, y] := Φ2(x, y)−Φ2(y, x).
I Lie Groups −→ Formal Group Laws ←→ Lie Algebras
I This correspondence is true over fields F of characteristic zero, andis lost if char(F) 6= 0. (See J.-P. Serre, Lie Algebras and Lie Groups,LNM 1500, Springer, 1992).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
I Viceversa, given a formal group we can construct a Lie algebra.
I Construction. Let us write
Φ(x, y) = x + y + Φ2(x, y) + Φ3(x, y) + . . .
where Φ2(x , y) denotes the quadratic part of the formal group law,Φ3(x , y) the cubic part, and so on.Any n-dimensional formal group law induces an n-dimensional Liealgebra over the same ring R, by means of the formula
[x, y] := Φ2(x, y)−Φ2(y, x).
I Lie Groups −→ Formal Group Laws ←→ Lie Algebras
I This correspondence is true over fields F of characteristic zero, andis lost if char(F) 6= 0. (See J.-P. Serre, Lie Algebras and Lie Groups,LNM 1500, Springer, 1992).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
I Viceversa, given a formal group we can construct a Lie algebra.
I Construction. Let us write
Φ(x, y) = x + y + Φ2(x, y) + Φ3(x, y) + . . .
where Φ2(x , y) denotes the quadratic part of the formal group law,Φ3(x , y) the cubic part, and so on.Any n-dimensional formal group law induces an n-dimensional Liealgebra over the same ring R, by means of the formula
[x, y] := Φ2(x, y)−Φ2(y, x).
I Lie Groups −→ Formal Group Laws ←→ Lie Algebras
I This correspondence is true over fields F of characteristic zero, andis lost if char(F) 6= 0. (See J.-P. Serre, Lie Algebras and Lie Groups,LNM 1500, Springer, 1992).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Examples
I The additive group law
Φ(x , y) = x + y
I The multiplicative group law
Φ(x , y) = x + y + axy
I The hyperbolic group law (addition of velocities in special relativity)
Φ(x , y) =x + y
1 + xy
I The Euler group law for elliptic integrals.
Φ(x , y) = (x√
1− y 4 + y√
1− x4)/(1 + x2y 2)
I ∫ x
0
dt√1− t4
+
∫ y
0
dt√1− t4
=
∫ Φ(x,y)
0
dt√1− t4
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Examples
I The additive group law
Φ(x , y) = x + y
I The multiplicative group law
Φ(x , y) = x + y + axy
I The hyperbolic group law (addition of velocities in special relativity)
Φ(x , y) =x + y
1 + xy
I The Euler group law for elliptic integrals.
Φ(x , y) = (x√
1− y 4 + y√
1− x4)/(1 + x2y 2)
I ∫ x
0
dt√1− t4
+
∫ y
0
dt√1− t4
=
∫ Φ(x,y)
0
dt√1− t4
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Examples
I The additive group law
Φ(x , y) = x + y
I The multiplicative group law
Φ(x , y) = x + y + axy
I The hyperbolic group law (addition of velocities in special relativity)
Φ(x , y) =x + y
1 + xy
I The Euler group law for elliptic integrals.
Φ(x , y) = (x√
1− y 4 + y√
1− x4)/(1 + x2y 2)
I ∫ x
0
dt√1− t4
+
∫ y
0
dt√1− t4
=
∫ Φ(x,y)
0
dt√1− t4
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Examples
I The additive group law
Φ(x , y) = x + y
I The multiplicative group law
Φ(x , y) = x + y + axy
I The hyperbolic group law (addition of velocities in special relativity)
Φ(x , y) =x + y
1 + xy
I The Euler group law for elliptic integrals.
Φ(x , y) = (x√
1− y 4 + y√
1− x4)/(1 + x2y 2)
I ∫ x
0
dt√1− t4
+
∫ y
0
dt√1− t4
=
∫ Φ(x,y)
0
dt√1− t4
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Examples
I The additive group law
Φ(x , y) = x + y
I The multiplicative group law
Φ(x , y) = x + y + axy
I The hyperbolic group law (addition of velocities in special relativity)
Φ(x , y) =x + y
1 + xy
I The Euler group law for elliptic integrals.
Φ(x , y) = (x√
1− y 4 + y√
1− x4)/(1 + x2y 2)
I ∫ x
0
dt√1− t4
+
∫ y
0
dt√1− t4
=
∫ Φ(x,y)
0
dt√1− t4
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Motivation: Thermodynamics
I The additive group law ⇐⇒ The composition law for the SBentropy:
Φ(x , y) = x + y
SBG (A ∪ B) = SBG (A) + SBG (B)
I The multiplicative group law ⇐⇒ The composition law for theTsallis entropy:
Φ(x , y) = x + y + axy
Sq(A ∪ B) = Sq(A) + Sq(B) + (1− q)Sq(A)Sq(B)
I These two results clearly support the existence of a directconnection among group laws and composition laws of generalizedentropies.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Motivation: Thermodynamics
I The additive group law ⇐⇒ The composition law for the SBentropy:
Φ(x , y) = x + y
SBG (A ∪ B) = SBG (A) + SBG (B)
I The multiplicative group law ⇐⇒ The composition law for theTsallis entropy:
Φ(x , y) = x + y + axy
Sq(A ∪ B) = Sq(A) + Sq(B) + (1− q)Sq(A)Sq(B)
I These two results clearly support the existence of a directconnection among group laws and composition laws of generalizedentropies.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Motivation: Thermodynamics
I The additive group law ⇐⇒ The composition law for the SBentropy:
Φ(x , y) = x + y
SBG (A ∪ B) = SBG (A) + SBG (B)
I The multiplicative group law ⇐⇒ The composition law for theTsallis entropy:
Φ(x , y) = x + y + axy
Sq(A ∪ B) = Sq(A) + Sq(B) + (1− q)Sq(A)Sq(B)
I These two results clearly support the existence of a directconnection among group laws and composition laws of generalizedentropies.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
A generalized rational group law (V. M. Bukhshtaber, A. N.Khodolov, Math. USSR Sbornik, 69, 77–97 (1991)).
I
Φ(x , y) =x + y + axy
1 + bxy
I A particular case of this group law (a = α− 1), b = α alreadyappeared in the context of the Hirzebruch theory of genera inalgebraic topology:
Φ(x , y) =x + y + (α− 1)xy
1 + αxy
(V. M. Bukhshtaber, A. S. Mishchenko, S. P. Novikov, Russ.Math. Surv. 26, 63–90 (1971)).
I When a = b = 0, we obtain the additive group law.I When b = 0, we obtain the multiplicative group law.I Also, when a = 0, b = 1, we obtain the hyperbolic group law.
(See also E. Curado, P. Tempesta, C. Tsallis, preprint arxiv:1507.05058, (2015)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
A generalized rational group law (V. M. Bukhshtaber, A. N.Khodolov, Math. USSR Sbornik, 69, 77–97 (1991)).
I
Φ(x , y) =x + y + axy
1 + bxy
I A particular case of this group law (a = α− 1), b = α alreadyappeared in the context of the Hirzebruch theory of genera inalgebraic topology:
Φ(x , y) =x + y + (α− 1)xy
1 + αxy
(V. M. Bukhshtaber, A. S. Mishchenko, S. P. Novikov, Russ.Math. Surv. 26, 63–90 (1971)).
I When a = b = 0, we obtain the additive group law.I When b = 0, we obtain the multiplicative group law.I Also, when a = 0, b = 1, we obtain the hyperbolic group law.
(See also E. Curado, P. Tempesta, C. Tsallis, preprint arxiv:1507.05058, (2015)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
A generalized rational group law (V. M. Bukhshtaber, A. N.Khodolov, Math. USSR Sbornik, 69, 77–97 (1991)).
I
Φ(x , y) =x + y + axy
1 + bxy
I A particular case of this group law (a = α− 1), b = α alreadyappeared in the context of the Hirzebruch theory of genera inalgebraic topology:
Φ(x , y) =x + y + (α− 1)xy
1 + αxy
(V. M. Bukhshtaber, A. S. Mishchenko, S. P. Novikov, Russ.Math. Surv. 26, 63–90 (1971)).
I When a = b = 0, we obtain the additive group law.
I When b = 0, we obtain the multiplicative group law.I Also, when a = 0, b = 1, we obtain the hyperbolic group law.
(See also E. Curado, P. Tempesta, C. Tsallis, preprint arxiv:1507.05058, (2015)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
A generalized rational group law (V. M. Bukhshtaber, A. N.Khodolov, Math. USSR Sbornik, 69, 77–97 (1991)).
I
Φ(x , y) =x + y + axy
1 + bxy
I A particular case of this group law (a = α− 1), b = α alreadyappeared in the context of the Hirzebruch theory of genera inalgebraic topology:
Φ(x , y) =x + y + (α− 1)xy
1 + αxy
(V. M. Bukhshtaber, A. S. Mishchenko, S. P. Novikov, Russ.Math. Surv. 26, 63–90 (1971)).
I When a = b = 0, we obtain the additive group law.I When b = 0, we obtain the multiplicative group law.
I Also, when a = 0, b = 1, we obtain the hyperbolic group law.(See also E. Curado, P. Tempesta, C. Tsallis, preprint arxiv:1507.05058, (2015)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
A generalized rational group law (V. M. Bukhshtaber, A. N.Khodolov, Math. USSR Sbornik, 69, 77–97 (1991)).
I
Φ(x , y) =x + y + axy
1 + bxy
I A particular case of this group law (a = α− 1), b = α alreadyappeared in the context of the Hirzebruch theory of genera inalgebraic topology:
Φ(x , y) =x + y + (α− 1)xy
1 + αxy
(V. M. Bukhshtaber, A. S. Mishchenko, S. P. Novikov, Russ.Math. Surv. 26, 63–90 (1971)).
I When a = b = 0, we obtain the additive group law.I When b = 0, we obtain the multiplicative group law.I Also, when a = 0, b = 1, we obtain the hyperbolic group law.
(See also E. Curado, P. Tempesta, C. Tsallis, preprint arxiv:1507.05058, (2015)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
The Lazard Universal Formal Group
Let us consider the series (the so called formal group exponential)
G (t) =∞∑k=0
aktk+1
k + 1(7)
over the ring B = Z[a1, a2, ...] of integral polynomials in infinitely manyvariables a1, a2, . . ., with a0 = 1. Let G−1 (s) be its compositional inverse:
G−1 (s) =∞∑i=0
bis i+1
i + 1, (8)
so that G−1 (G (t)) = t. We have b0 = 1, b1 = −a1, b2 = 32 a2
1 − a2, . . .(Lagrange theorem). The Lazard universal formal group law is definedto be the formal power series
Φ (x, y) = G(G−1 (x) + G−1 (y)
).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
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Properties
I The following two important results, due to Lazard, hold.
I Proposition 1.8. For any commutative one-dimensional formalgroup law over any ring R, there exists a unique homomorphismL→ R under which the Lazard group law is mapped into the givengroup law (the so called universal property of the Lazard group).
I Proposition 1.9. For any commutative one-dimensional formalgroup law Ψ(x , y) over any ring R, there exists a seriesψ(x) ∈ R[[x ]]⊗Q such that
ψ(x) = x + O(x2), and Ψ(x , y) = ψ−1 (ψ(x) + ψ(y)) ∈ R[[x , y ]]⊗Q.
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Introduction: A Foundational PerspectiveFormal Group Theory
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DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Properties
I The following two important results, due to Lazard, hold.
I Proposition 1.8. For any commutative one-dimensional formalgroup law over any ring R, there exists a unique homomorphismL→ R under which the Lazard group law is mapped into the givengroup law (the so called universal property of the Lazard group).
I Proposition 1.9. For any commutative one-dimensional formalgroup law Ψ(x , y) over any ring R, there exists a seriesψ(x) ∈ R[[x ]]⊗Q such that
ψ(x) = x + O(x2), and Ψ(x , y) = ψ−1 (ψ(x) + ψ(y)) ∈ R[[x , y ]]⊗Q.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
Properties
I The following two important results, due to Lazard, hold.
I Proposition 1.8. For any commutative one-dimensional formalgroup law over any ring R, there exists a unique homomorphismL→ R under which the Lazard group law is mapped into the givengroup law (the so called universal property of the Lazard group).
I Proposition 1.9. For any commutative one-dimensional formalgroup law Ψ(x , y) over any ring R, there exists a seriesψ(x) ∈ R[[x ]]⊗Q such that
ψ(x) = x + O(x2), and Ψ(x , y) = ψ−1 (ψ(x) + ψ(y)) ∈ R[[x , y ]]⊗Q.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
The cohomological interpretation of formal group theory(U can sleep here)
(Novikov’s school, Quillen, Ray, etc.)There is a clear relationship between cobordism theory and formalgroup laws.Definition 1.10. Suppose we are assigned with some class of manifolds,closed and with boundary, and possibly with additional structure. Werequire thata) The boundary of a manifold in the class belongs to the same class;b) The direct product of manifolds of the class belong to the class(“multiplicative property”);c) a closed domain with smooth boundary in a manifold of the classbelongs to the class (“excision axiom”)We say that such a class defines a cobordism (and a bordism) theory,denoted by C.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
U can sleep here. Part II
Let M be a closed manifold in C. The cycles (bordisms in C) for anycomplex K are the pairs (M, f ) where f : M → K is a continuousmapping.The “films” are the pairs (N, g) where N ∈ C has boundary andg : N → K .Definition. The bordism group of K relative to the class C, denoted byΩCn (K ) is
ΩCn (K ) =group of n-dimensional cycles
group of boundaries of films in CFor finite complexes K , the Cobordism groups Ωn
C associated can beobtained by means of the Alexander-Pontryagin duality law. Let N ∈ Nlarge. If K ⊂ SN , where SN is a sphere, by definition we put
ΩnC(K ) = ΩCN−n
(SN ,SN \ K
)and this definition is independent of N and the embedding K ⊂ SN .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
U can sleep here. Part III
There are various correlated aspects of the theory:
I Unitary cobordisms
I Geometric bordisms
I Chern characters and Hirzebruch genera
I All the fundamental concepts and facts of the theory of unitarycobordisms, both modern and classical, can be expressed by means ofLazard’s formal group (S. P. Novikov).
I Main reason : if [CPn] are cobordism classes of complex projective spaces,the formal group exponential
G(u) =∑k≥0
[CPn]
n + 1un+1
generates the formal groups associated with the previous theories (byintroducing a suitable multiplication in each theory). At the same time,
G(G−1(u) + G−1(v))
generates the Lazard formal group.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
U can sleep here. Part III
There are various correlated aspects of the theory:
I Unitary cobordisms
I Geometric bordisms
I Chern characters and Hirzebruch genera
I All the fundamental concepts and facts of the theory of unitarycobordisms, both modern and classical, can be expressed by means ofLazard’s formal group (S. P. Novikov).
I Main reason : if [CPn] are cobordism classes of complex projective spaces,the formal group exponential
G(u) =∑k≥0
[CPn]
n + 1un+1
generates the formal groups associated with the previous theories (byintroducing a suitable multiplication in each theory). At the same time,
G(G−1(u) + G−1(v))
generates the Lazard formal group.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
U can sleep here. Part III
There are various correlated aspects of the theory:
I Unitary cobordisms
I Geometric bordisms
I Chern characters and Hirzebruch genera
I All the fundamental concepts and facts of the theory of unitarycobordisms, both modern and classical, can be expressed by means ofLazard’s formal group (S. P. Novikov).
I Main reason : if [CPn] are cobordism classes of complex projective spaces,the formal group exponential
G(u) =∑k≥0
[CPn]
n + 1un+1
generates the formal groups associated with the previous theories (byintroducing a suitable multiplication in each theory). At the same time,
G(G−1(u) + G−1(v))
generates the Lazard formal group.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
U can sleep here. Part III
There are various correlated aspects of the theory:
I Unitary cobordisms
I Geometric bordisms
I Chern characters and Hirzebruch genera
I All the fundamental concepts and facts of the theory of unitarycobordisms, both modern and classical, can be expressed by means ofLazard’s formal group (S. P. Novikov).
I Main reason : if [CPn] are cobordism classes of complex projective spaces,the formal group exponential
G(u) =∑k≥0
[CPn]
n + 1un+1
generates the formal groups associated with the previous theories (byintroducing a suitable multiplication in each theory). At the same time,
G(G−1(u) + G−1(v))
generates the Lazard formal group.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
DefinitionsExamplesA rational group lawThe Lazard Universal Formal GroupFormal Groups and Cobordism Theory
U can sleep here. Part III
There are various correlated aspects of the theory:
I Unitary cobordisms
I Geometric bordisms
I Chern characters and Hirzebruch genera
I All the fundamental concepts and facts of the theory of unitarycobordisms, both modern and classical, can be expressed by means ofLazard’s formal group (S. P. Novikov).
I Main reason : if [CPn] are cobordism classes of complex projective spaces,the formal group exponential
G(u) =∑k≥0
[CPn]
n + 1un+1
generates the formal groups associated with the previous theories (byintroducing a suitable multiplication in each theory). At the same time,
G(G−1(u) + G−1(v))
generates the Lazard formal group.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
Resume
The first three Shannon-Khinchin axioms, jointly with thecomposability axiom, are a necessary set of axioms to doThermodynamics.In particular, the composability axiom replaces and generalizes thefourth axiom, and provides a group-theoretical structure to the notion ofentropy.
I Definition 1.11. We shall say that an entropy isthermodynamically admissible if satisfies the first three SK axiomsand is composable.
I Every “decent” entropy has associated a group structure.
I Vice-versa, given a group structure, is there a suitable entropyassociated with it? (Lectures II and III).
I In particular, is there any mathematical obstruction to thiscorrespondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
Resume
The first three Shannon-Khinchin axioms, jointly with thecomposability axiom, are a necessary set of axioms to doThermodynamics.In particular, the composability axiom replaces and generalizes thefourth axiom, and provides a group-theoretical structure to the notion ofentropy.
I Definition 1.11. We shall say that an entropy isthermodynamically admissible if satisfies the first three SK axiomsand is composable.
I Every “decent” entropy has associated a group structure.
I Vice-versa, given a group structure, is there a suitable entropyassociated with it? (Lectures II and III).
I In particular, is there any mathematical obstruction to thiscorrespondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
Resume
The first three Shannon-Khinchin axioms, jointly with thecomposability axiom, are a necessary set of axioms to doThermodynamics.In particular, the composability axiom replaces and generalizes thefourth axiom, and provides a group-theoretical structure to the notion ofentropy.
I Definition 1.11. We shall say that an entropy isthermodynamically admissible if satisfies the first three SK axiomsand is composable.
I Every “decent” entropy has associated a group structure.
I Vice-versa, given a group structure, is there a suitable entropyassociated with it? (Lectures II and III).
I In particular, is there any mathematical obstruction to thiscorrespondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Resume
The first three Shannon-Khinchin axioms, jointly with thecomposability axiom, are a necessary set of axioms to doThermodynamics.In particular, the composability axiom replaces and generalizes thefourth axiom, and provides a group-theoretical structure to the notion ofentropy.
I Definition 1.11. We shall say that an entropy isthermodynamically admissible if satisfies the first three SK axiomsand is composable.
I Every “decent” entropy has associated a group structure.
I Vice-versa, given a group structure, is there a suitable entropyassociated with it? (Lectures II and III).
I In particular, is there any mathematical obstruction to thiscorrespondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
Another uniqueness theorem
P. T., Formal groups and Z-entropies, arxiv: 1507.07436 (2015).Let us denote by N the number of particles of a complex system.Definition 1.12. An entropy S is said to be extensive if there exists a function(“occupation law”) W = W (N) such that, over the uniform distributionpi = 1/W , i = 1, . . . ,W , we have, for large values of N:
S [W (N)] ∝ N
Observation. No entropy is extensive in all possible regimes. For instance, theBoltzmann entropy is extensive when the occupation law of the phase space isW (N) = kN ; the Tsallis entropy is extensive for W (N) = Nρ, and so on.Definition 1.13. An entropy belongs to the trace-form class if it can bewritten in the general form S [p] =
∑Wi=1 F (pi ), where F : [0, 1] −→ R+ ∪ 0 is
assumed to be a function at least twice differentiable in (0, 1), strictly concave,with the further properties F (0) = F (1) = 0. We shall also assume that S beextensive. The previous definition ensures that S [p] satisfies the axioms(SK1)-(SK3).
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General properties of the trace-form class
I Theorem 1.11 The Boltzmann entropy and the Tsallis entropy arethe only trace-form and C∞(0, 1) entropies which are strictlycomposable.
I Consequence. The trace-form class has a serious “drawback”.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
I Theorem 1.11 The Boltzmann entropy and the Tsallis entropy arethe only trace-form and C∞(0, 1) entropies which are strictlycomposable.
I Consequence. The trace-form class has a serious “drawback”.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
Strong and weak composability
I The composability axiom is very demanding: for trace-formentropies
∑Wi=1 f (pi ), only two entropies are strictly composable:
The entropy SBG =∑
i pi ln 1p1
, the entropy Sq =∑
i pqi −1
1−q .
I We can consider a weaker formulation of this notion.
I Definition 2. An entropy is weakly composable if the properties(C1)–(C3) of the composability axiom are satisfied at least when theprobability distributions of the two independent subsystems A and Bare both uniform, and property (C4) holds in general.
I The uniform distribution emerges when considering isolated physicalsystems at the equilibrium (microcanonical ensemble), or incontact with a thermostat at very high temperature (canonicalensemble) : for instance, for high-temperature astrophysical objects.A large set of entropies recently considered are weakly composable,although even not all of them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Strong and weak composability
I The composability axiom is very demanding: for trace-formentropies
∑Wi=1 f (pi ), only two entropies are strictly composable:
The entropy SBG =∑
i pi ln 1p1
, the entropy Sq =∑
i pqi −1
1−q .I We can consider a weaker formulation of this notion.
I Definition 2. An entropy is weakly composable if the properties(C1)–(C3) of the composability axiom are satisfied at least when theprobability distributions of the two independent subsystems A and Bare both uniform, and property (C4) holds in general.
I The uniform distribution emerges when considering isolated physicalsystems at the equilibrium (microcanonical ensemble), or incontact with a thermostat at very high temperature (canonicalensemble) : for instance, for high-temperature astrophysical objects.A large set of entropies recently considered are weakly composable,although even not all of them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Strong and weak composability
I The composability axiom is very demanding: for trace-formentropies
∑Wi=1 f (pi ), only two entropies are strictly composable:
The entropy SBG =∑
i pi ln 1p1
, the entropy Sq =∑
i pqi −1
1−q .I We can consider a weaker formulation of this notion.
I Definition 2. An entropy is weakly composable if the properties(C1)–(C3) of the composability axiom are satisfied at least when theprobability distributions of the two independent subsystems A and Bare both uniform, and property (C4) holds in general.
I The uniform distribution emerges when considering isolated physicalsystems at the equilibrium (microcanonical ensemble), or incontact with a thermostat at very high temperature (canonicalensemble) : for instance, for high-temperature astrophysical objects.A large set of entropies recently considered are weakly composable,although even not all of them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Strong and weak composability
I The composability axiom is very demanding: for trace-formentropies
∑Wi=1 f (pi ), only two entropies are strictly composable:
The entropy SBG =∑
i pi ln 1p1
, the entropy Sq =∑
i pqi −1
1−q .I We can consider a weaker formulation of this notion.
I Definition 2. An entropy is weakly composable if the properties(C1)–(C3) of the composability axiom are satisfied at least when theprobability distributions of the two independent subsystems A and Bare both uniform, and property (C4) holds in general.
I The uniform distribution emerges when considering isolated physicalsystems at the equilibrium (microcanonical ensemble), or incontact with a thermostat at very high temperature (canonicalensemble) : for instance, for high-temperature astrophysical objects.A large set of entropies recently considered are weakly composable,although even not all of them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Examples of weakly composable entropies
Typical examples of this (huge) class are the most well-known entropies.
I The Kaniadakis entropy
Sκ[p] = kB
W∑i=1
pip−κi − pκi
2κ(9)
I The Anteneodo-Plastino entropy
Sη =W∑i=1
(Γ
(η + 1
η,− ln pi
)− piΓ
(η + 1
η
)).
I The HT Sc,d entropy
Sc,d =e
1− c + cd
W∑i=1
Γ(1 + d , 1− c ln pi )−c
1− c + cd. (10)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Examples of weakly composable entropies
Typical examples of this (huge) class are the most well-known entropies.
I The Kaniadakis entropy
Sκ[p] = kB
W∑i=1
pip−κi − pκi
2κ(9)
I The Anteneodo-Plastino entropy
Sη =W∑i=1
(Γ
(η + 1
η,− ln pi
)− piΓ
(η + 1
η
)).
I The HT Sc,d entropy
Sc,d =e
1− c + cd
W∑i=1
Γ(1 + d , 1− c ln pi )−c
1− c + cd. (10)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
Examples of weakly composable entropies
Typical examples of this (huge) class are the most well-known entropies.
I The Kaniadakis entropy
Sκ[p] = kB
W∑i=1
pip−κi − pκi
2κ(9)
I The Anteneodo-Plastino entropy
Sη =W∑i=1
(Γ
(η + 1
η,− ln pi
)− piΓ
(η + 1
η
)).
I The HT Sc,d entropy
Sc,d =e
1− c + cd
W∑i=1
Γ(1 + d , 1− c ln pi )−c
1− c + cd. (10)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
I The Abe entropy
SAbe =W∑i=1
pq−1
i − pqi
q − q−1
I The Borges-Roditi entropy
SBR =W∑i=1
pq′
i − pqi
q − q′
I The Tsallis-Cirto entropy
Sδ = kB
W∑i=1
pi
(ln
1
pi
)δ, 0 < δ ≤ (1 + ln W ). (11)
I The Shafee entropy
Sδ = kB
W∑i=1
pci
(ln
1
pi
). (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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General properties of the trace-form class
II The Abe entropy
SAbe =W∑i=1
pq−1
i − pqi
q − q−1
I The Borges-Roditi entropy
SBR =W∑i=1
pq′
i − pqi
q − q′
I The Tsallis-Cirto entropy
Sδ = kB
W∑i=1
pi
(ln
1
pi
)δ, 0 < δ ≤ (1 + ln W ). (11)
I The Shafee entropy
Sδ = kB
W∑i=1
pci
(ln
1
pi
). (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
II The Abe entropy
SAbe =W∑i=1
pq−1
i − pqi
q − q−1
I The Borges-Roditi entropy
SBR =W∑i=1
pq′
i − pqi
q − q′
I The Tsallis-Cirto entropy
Sδ = kB
W∑i=1
pi
(ln
1
pi
)δ, 0 < δ ≤ (1 + ln W ). (11)
I The Shafee entropy
Sδ = kB
W∑i=1
pci
(ln
1
pi
). (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
General properties of the trace-form class
II The Abe entropy
SAbe =W∑i=1
pq−1
i − pqi
q − q−1
I The Borges-Roditi entropy
SBR =W∑i=1
pq′
i − pqi
q − q′
I The Tsallis-Cirto entropy
Sδ = kB
W∑i=1
pi
(ln
1
pi
)δ, 0 < δ ≤ (1 + ln W ). (11)
I The Shafee entropy
Sδ = kB
W∑i=1
pci
(ln
1
pi
). (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
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Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Entropies and the discrete world
II There is a quite interesting and unexpected relationship between thetheory of difference equations and a large class of generalizedentropies.
Main idea: the nonstandard logarithm appearing at the heart of thenotion of generalized entropy can be seen (in most cases) as arealization of a specific difference operator.
I The modern algebraic theory of difference operators is intimatelyrelated with the theory of polynomial sequences of Sheffer andAppell type.
It is due to G. C. Rota and collaborators (finite operator theory,umbral calculus).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Entropies and the discrete world
I There is a quite interesting and unexpected relationship between thetheory of difference equations and a large class of generalizedentropies.
Main idea: the nonstandard logarithm appearing at the heart of thenotion of generalized entropy can be seen (in most cases) as arealization of a specific difference operator.
I The modern algebraic theory of difference operators is intimatelyrelated with the theory of polynomial sequences of Sheffer andAppell type.
It is due to G. C. Rota and collaborators (finite operator theory,umbral calculus).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I In the following, we shall consider a regular equally spaced latticeof points, denoted by L and indexed by x = nσ, with n ∈ N, σ ∈ R.
Also, we shall deal with operators acting on the algebra of(sufficiently regular) functions FL defined on L.
I The crucial notion of the theory is that of Delta Operator. Todefine it, we shall first introduce shift operators.
I The shift operator, denoted by T , is the operator on L, whoseaction on a function f ∈ FL is given by
Tf (x) = f (x + σ) .
As usual, the operator T can also be represented in terms ofdifferential operators as T = eσD ≡ eσt .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I In the following, we shall consider a regular equally spaced latticeof points, denoted by L and indexed by x = nσ, with n ∈ N, σ ∈ R.
Also, we shall deal with operators acting on the algebra of(sufficiently regular) functions FL defined on L.
I The crucial notion of the theory is that of Delta Operator. Todefine it, we shall first introduce shift operators.
I The shift operator, denoted by T , is the operator on L, whoseaction on a function f ∈ FL is given by
Tf (x) = f (x + σ) .
As usual, the operator T can also be represented in terms ofdifferential operators as T = eσD ≡ eσt .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I In the following, we shall consider a regular equally spaced latticeof points, denoted by L and indexed by x = nσ, with n ∈ N, σ ∈ R.
Also, we shall deal with operators acting on the algebra of(sufficiently regular) functions FL defined on L.
I The crucial notion of the theory is that of Delta Operator. Todefine it, we shall first introduce shift operators.
I The shift operator, denoted by T , is the operator on L, whoseaction on a function f ∈ FL is given by
Tf (x) = f (x + σ) .
As usual, the operator T can also be represented in terms ofdifferential operators as T = eσD ≡ eσt .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I Definition 1.14. An operator S is said to be shift–invariant if commuteswith T : [S ,T ] = 0.
I Definition 1.15. A delta operator Q is a shift–invariant operator suchthat Qx = const 6= 0.
Directly from this definition we deduce the following property.
I Corollary For every constant c ∈ R, Qc = 0.
I Examples. The most common example of delta operator are provided by
a) the standard derivative: Q = ∂x .
b) The forward discrete derivative
Q = ∆+ :=T − 1
σ
c) The backward discrete derivative
Q = ∆− :=1− T−1
σ
d) The symmetric discrete derivative
Q = ∆s :=T − T−1
2σ
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I Definition 1.14. An operator S is said to be shift–invariant if commuteswith T : [S ,T ] = 0.
I Definition 1.15. A delta operator Q is a shift–invariant operator suchthat Qx = const 6= 0.
Directly from this definition we deduce the following property.
I Corollary For every constant c ∈ R, Qc = 0.
I Examples. The most common example of delta operator are provided by
a) the standard derivative: Q = ∂x .
b) The forward discrete derivative
Q = ∆+ :=T − 1
σ
c) The backward discrete derivative
Q = ∆− :=1− T−1
σ
d) The symmetric discrete derivative
Q = ∆s :=T − T−1
2σ
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I Definition 1.14. An operator S is said to be shift–invariant if commuteswith T : [S ,T ] = 0.
I Definition 1.15. A delta operator Q is a shift–invariant operator suchthat Qx = const 6= 0.
Directly from this definition we deduce the following property.
I Corollary For every constant c ∈ R, Qc = 0.
I Examples. The most common example of delta operator are provided by
a) the standard derivative: Q = ∂x .
b) The forward discrete derivative
Q = ∆+ :=T − 1
σ
c) The backward discrete derivative
Q = ∆− :=1− T−1
σ
d) The symmetric discrete derivative
Q = ∆s :=T − T−1
2σ
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I Definition 1.14. An operator S is said to be shift–invariant if commuteswith T : [S ,T ] = 0.
I Definition 1.15. A delta operator Q is a shift–invariant operator suchthat Qx = const 6= 0.
Directly from this definition we deduce the following property.
I Corollary For every constant c ∈ R, Qc = 0.
I Examples. The most common example of delta operator are provided by
a) the standard derivative: Q = ∂x .
b) The forward discrete derivative
Q = ∆+ :=T − 1
σ
c) The backward discrete derivative
Q = ∆− :=1− T−1
σ
d) The symmetric discrete derivative
Q = ∆s :=T − T−1
2σ
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I Definition 1.16.Given a delta operator Q, a polynomial sequence pn (x)n∈N willbe said to be the sequence of basic polynomials for Q if thefollowing conditions are satisfied:
1) p0 (x) = 1;
2) pn (0) = 0 for all n > 0;
3) Qpn (x) = npn−1 (x) .
Notice that, for a given delta operator Q the sequence of associatedbasic polynomials is unique.
I Examples.Let Q = ∂x . We have pn(x) = xn
Qxn = nxn−1
and the other properties are obvious.Let Q = ∆+. The basic polynomials for Q are the lower factorialpolynomials (x)n := x(x − 1) · . . . · (x − n + 1). We have
∆+(x)n = n(x)n−1.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I Definition 1.16.Given a delta operator Q, a polynomial sequence pn (x)n∈N willbe said to be the sequence of basic polynomials for Q if thefollowing conditions are satisfied:
1) p0 (x) = 1;
2) pn (0) = 0 for all n > 0;
3) Qpn (x) = npn−1 (x) .
Notice that, for a given delta operator Q the sequence of associatedbasic polynomials is unique.
I Examples.Let Q = ∂x . We have pn(x) = xn
Qxn = nxn−1
and the other properties are obvious.Let Q = ∆+. The basic polynomials for Q are the lower factorialpolynomials (x)n := x(x − 1) · . . . · (x − n + 1). We have
∆+(x)n = n(x)n−1.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I As has been proved by G. C. Rota, there is an isomorphism between thering of formal power series in a variable t and the ring of shift–invariantoperators, carrying
f (t) =∞∑k=0
aktk
k!into
∞∑k=0
akQk
k!. (13)
Definition 1.17. Given a delta operator Q, the function (or a formalpower series) G (t), associated with S under Rota’s isomorphism will becalled the indicator of S .
I With a slight abuse of notation, in the following we often identify a deltaoperator with its indicator. For instance, for ∆+ we have:
∆+ =T − 1
σ=
eσ∂x − 1
σ=∞∑k=0
(σ)k−1∂kx
k!⇐⇒ δ+(t) =
eσt − 1
σ.
For ∆s , we deduce
∆s =T − T−1
2σ=⇐⇒ δs(t) =
eσt − s−σt
2σ.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
I As has been proved by G. C. Rota, there is an isomorphism between thering of formal power series in a variable t and the ring of shift–invariantoperators, carrying
f (t) =∞∑k=0
aktk
k!into
∞∑k=0
akQk
k!. (13)
Definition 1.17. Given a delta operator Q, the function (or a formalpower series) G (t), associated with S under Rota’s isomorphism will becalled the indicator of S .
I With a slight abuse of notation, in the following we often identify a deltaoperator with its indicator. For instance, for ∆+ we have:
∆+ =T − 1
σ=
eσ∂x − 1
σ=∞∑k=0
(σ)k−1∂kx
k!⇐⇒ δ+(t) =
eσt − 1
σ.
For ∆s , we deduce
∆s =T − T−1
2σ=⇐⇒ δs(t) =
eσt − s−σt
2σ.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Definition 1.17 An Appell sequence of polynomials is a Sheffersequence for the delta operator ∂x .
Consequence. For an Appell of polynomials pn (x), we have
∂xpn (x) = npn−1 (x) .
Among the most important examples of Appell sequences, apart fromxnn∈N, are the classical Bernoulli and Euler polynomials and theirgeneralizations.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Difference operators and generalized entropies
In order to define the class of entropic functionals of interest for thiswork, we will consider operators expressed as finite Laurent series in shiftoperators
∆r =1
σ
m∑n=l
knT n, l , m ∈ Z, m − l = r > 0, (14)
where kn are real constants such that
m∑n=l
kn = 0,m∑n=l
nkn = c . (15)
and km 6= 0, kl 6= 0. We choose c = 1, to reproduce the derivative ∂x inthe continuum limit, when the lattice spacing σ → 0. The main ideaunderlying our construction is to represent delta operators in logarithmicform, in terms of a suitable function.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Generalized logarithms from delta operators
I Definition 1.18. We call logarithmic representation of the delta operator(14) the correspondence T ↔ eσ ln x ≡ xσ, that defines an isomorphism Ibetween the space of shift–invariant operators and the space of functionsf ∈ H.
Definition 1.19. We call generalized logarithm the function
LogG (x) := G(ln x) =1
σ
m∑n=l
knxσn, l ,m ∈ Z,
l < m, m − l = r , x > 0 (16)
with the constraints (15), i.e. the image of the operator (14) under theisomorphism I.
I Lemma 1.20 The following property holds:
limσ→0
LogG (x) = ln x . (17)
Tsallis’ logarithm is
logq(x) =x1−q − 1
1− q= G(ln x), G(t) = δ+(t) =
e(1−q)t − 1
1− q
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Generalized logarithms from delta operators
I Definition 1.18. We call logarithmic representation of the delta operator(14) the correspondence T ↔ eσ ln x ≡ xσ, that defines an isomorphism Ibetween the space of shift–invariant operators and the space of functionsf ∈ H.
Definition 1.19. We call generalized logarithm the function
LogG (x) := G(ln x) =1
σ
m∑n=l
knxσn, l ,m ∈ Z,
l < m, m − l = r , x > 0 (16)
with the constraints (15), i.e. the image of the operator (14) under theisomorphism I.
I Lemma 1.20 The following property holds:
limσ→0
LogG (x) = ln x . (17)
Tsallis’ logarithm is
logq(x) =x1−q − 1
1− q= G(ln x), G(t) = δ+(t) =
e(1−q)t − 1
1− qPiergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Other examples
I Kaniadakis’ logarithm is
logk(x) =xk − x−k
2k= G(ln x), G(t) = δs(t) =
ekt − e−kt
2k.
Let us now construct some new examples. Consider for instance thedifference operators
∆III = T−2T−1+T−2
σ, ∆IV =
T 2− 32T+ 3
2T−1−T−2
σ,
∆V = T 3−2T 2+2T−2T−1+T−2
−σ ,
and so on.
I The corresponding generalized logarithms are
LogGIII (x) =1
σ
(xσ − 2x−σ + x−2σ
),
LogGIV (x) =1
σ
(x2σ − 3
2xσ +
3
2x−σ − x−2σ
),
LogGV (x) =1
σ
(x3σ − 2x2σ + 2xσ − 2x−σ + x−2σ
).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Other examples
I Kaniadakis’ logarithm is
logk(x) =xk − x−k
2k= G(ln x), G(t) = δs(t) =
ekt − e−kt
2k.
Let us now construct some new examples. Consider for instance thedifference operators
∆III = T−2T−1+T−2
σ, ∆IV =
T 2− 32T+ 3
2T−1−T−2
σ,
∆V = T 3−2T 2+2T−2T−1+T−2
−σ ,
and so on.
I The corresponding generalized logarithms are
LogGIII (x) =1
σ
(xσ − 2x−σ + x−2σ
),
LogGIV (x) =1
σ
(x2σ − 3
2xσ +
3
2x−σ − x−2σ
),
LogGV (x) =1
σ
(x3σ − 2x2σ + 2xσ − 2x−σ + x−2σ
).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
State of Art
I A generalization of the Shannon-Khinchin axiomatic formulation,originally proposed to characterize uniquely the Boltzmann entropy,has been proposed.
I Although the first three SK axioms are mantained, the fourth one isreplaced by the new composability axiom.
I This axiom encodes a group theoretical structure, which isresponsible of essentially all the properties of a generalized entropy.
I Formal group theory, borrowed from algebraic topology, has beenused to characterize the composability process, and consequently,the class of thermodynamically admissible entropies.
I A connection between discrete mathematics and generalizedlogarithms has been sketched, via the finite operator theory by C.G. Rota and coll.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
State of Art
I A generalization of the Shannon-Khinchin axiomatic formulation,originally proposed to characterize uniquely the Boltzmann entropy,has been proposed.
I Although the first three SK axioms are mantained, the fourth one isreplaced by the new composability axiom.
I This axiom encodes a group theoretical structure, which isresponsible of essentially all the properties of a generalized entropy.
I Formal group theory, borrowed from algebraic topology, has beenused to characterize the composability process, and consequently,the class of thermodynamically admissible entropies.
I A connection between discrete mathematics and generalizedlogarithms has been sketched, via the finite operator theory by C.G. Rota and coll.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
State of Art
I A generalization of the Shannon-Khinchin axiomatic formulation,originally proposed to characterize uniquely the Boltzmann entropy,has been proposed.
I Although the first three SK axioms are mantained, the fourth one isreplaced by the new composability axiom.
I This axiom encodes a group theoretical structure, which isresponsible of essentially all the properties of a generalized entropy.
I Formal group theory, borrowed from algebraic topology, has beenused to characterize the composability process, and consequently,the class of thermodynamically admissible entropies.
I A connection between discrete mathematics and generalizedlogarithms has been sketched, via the finite operator theory by C.G. Rota and coll.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
State of Art
I A generalization of the Shannon-Khinchin axiomatic formulation,originally proposed to characterize uniquely the Boltzmann entropy,has been proposed.
I Although the first three SK axioms are mantained, the fourth one isreplaced by the new composability axiom.
I This axiom encodes a group theoretical structure, which isresponsible of essentially all the properties of a generalized entropy.
I Formal group theory, borrowed from algebraic topology, has beenused to characterize the composability process, and consequently,the class of thermodynamically admissible entropies.
I A connection between discrete mathematics and generalizedlogarithms has been sketched, via the finite operator theory by C.G. Rota and coll.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
State of Art
I A generalization of the Shannon-Khinchin axiomatic formulation,originally proposed to characterize uniquely the Boltzmann entropy,has been proposed.
I Although the first three SK axioms are mantained, the fourth one isreplaced by the new composability axiom.
I This axiom encodes a group theoretical structure, which isresponsible of essentially all the properties of a generalized entropy.
I Formal group theory, borrowed from algebraic topology, has beenused to characterize the composability process, and consequently,the class of thermodynamically admissible entropies.
I A connection between discrete mathematics and generalizedlogarithms has been sketched, via the finite operator theory by C.G. Rota and coll.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
SK and
Composability
axioms Generalized Entropies
Group-theoretical
Structure
Delta Operators Formal group
theory
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
Introduction: A Foundational PerspectiveFormal Group Theory
Strong and weak composabilityGroups, Difference Operators and Entropies
Delta operators: a short reviewBasic sequences, Sheffer and Appell sequencesGeneralized logarithms from delta operators
Thank you!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT LECTURES ON GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
GROUPS, ENTROPIES AND NUMBERTHEORY
Piergiulio Tempesta
Universidad Complutense de Madridand
Instituto de Ciencias Matematicas (ICMAT),Madrid, Spain.
TEMPLETON SCHOOL ON FOUNDATIONS OF COMPLEXITY
October 14 - 15, 2015
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
LECTURE II
From Groups to Entropies: The composabilityaxiom and the Universal Group Entropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
Outline
The Universal Group EntropyThe universal-group entropyMain properties of SU entropyRelation with other entropies
Distribution functions and thermodynamic properties
An entropy for the rational group lawA new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A “magical recipe” to generate multiparametric entropiesThe microcanonical description and associated thermodynamicsMultiparametric entropies
On the asymptotic behaviour of generalized entropies
State of Art
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A little reminder: The composability axiom
An entropy S is (strictly) composable if there exists a continuous function oftwo real variables Φ(x , y) such that (C1)
S(A ∪ B) = Φ(S(A), S(B); η) (1)
where A ⊂ X and B ⊂ X are two statistically independent subsystems of agiven system X , defined over any probability distribution piWi=1, with thefurther properties(C2) Symmetry:
Φ(x , y) = Φ(y , x). (2)
(C3) Associativity:Φ(x ,Φ(y , z)) = Φ(Φ(x , y), z). (3)
(C4) Null-composability:Φ(x , 0) = x . (4)
An entropy is weakly composable if the properties (C1)–(C3) are satisfied atleast when the probability distributions of the two independent systems A andB are both uniform, and property (C4) holds in general.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The universal-group entropy
I Is there an entropy associated with the Lazard universal formalgroup?
I Let us denote by P the set of all probability distributions.Definition 2.1. Let pii=1,··· ,W , W ≥ 1, with
∑Wi=1 pi = 1, be a
discrete probability distribution. Let
G (t) =∞∑k=0
aktk+1
k + 1, (5)
be a real analytic function, where akk∈N is a sequence ofparameters, with a0 6= 0, such that the function SU : P → R+ ∪ 0,
SU(p1, . . . , pW ) := kB
W∑i=1
pi G
(ln
1
pi
), (6)
is a concave one. This function is the universal-group entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The universal-group entropy
I Is there an entropy associated with the Lazard universal formalgroup?
I Let us denote by P the set of all probability distributions.Definition 2.1. Let pii=1,··· ,W , W ≥ 1, with
∑Wi=1 pi = 1, be a
discrete probability distribution. Let
G (t) =∞∑k=0
aktk+1
k + 1, (5)
be a real analytic function, where akk∈N is a sequence ofparameters, with a0 6= 0, such that the function SU : P → R+ ∪ 0,
SU(p1, . . . , pW ) := kB
W∑i=1
pi G
(ln
1
pi
), (6)
is a concave one. This function is the universal-group entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
First properties
I Remark 2.2. The condition
ak > (k + 1)ak+1 ∀k ∈ N with akk∈N ≥ 0 (7)
is sufficient to ensure that the series G (t) is absolutely anduniformly convergent with a radius r =∞ and that SU [p] is astrictly concave functional. Although certainly restrictive, condition(19) is satisfied by many of the entropies known in the literature.
I Remark 2.3. The universal-group entropy depends on the infiniteset of parameters ak , which are a priori independent. To recoverknown cases of one-parametric or two-parametric entropies,depending for instance on the parameters q1 and q2, we shall havethat ak = fk(q1, q2).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
First properties
I Remark 2.2. The condition
ak > (k + 1)ak+1 ∀k ∈ N with akk∈N ≥ 0 (7)
is sufficient to ensure that the series G (t) is absolutely anduniformly convergent with a radius r =∞ and that SU [p] is astrictly concave functional. Although certainly restrictive, condition(19) is satisfied by many of the entropies known in the literature.
I Remark 2.3. The universal-group entropy depends on the infiniteset of parameters ak , which are a priori independent. To recoverknown cases of one-parametric or two-parametric entropies,depending for instance on the parameters q1 and q2, we shall havethat ak = fk(q1, q2).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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The universal-group entropyMain properties of SU entropyRelation with other entropies
Main properties of SU entropy
I Theorem 2.4 The entropy SU [p] is at least weakly composable.
I Proof. Let pAi WA
i=1 and pBj WB
j=1 two sets of probabilities associatedwith two statistically independent systems A and B. The jointprobability is given by
pA∪Bij = pAi · pBjThe total number of states of the composed system (including stateswith possibly null probability), is WAB = WAWB . We have
SU(A ∪ B) := kB
WA∑i=1
WB∑j=1
pA∪Bij G
(ln
1
pA∪Bij
)=
= kB
WA∑i=1
WB∑j=1
pAi ·pBj G
(ln
1
pAi+ ln
1
pBj
)= kB
WA∑i=1
WB∑j=1
pAi ·pBj G (t1 + t2) =
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Main properties of SU entropy
I Theorem 2.4 The entropy SU [p] is at least weakly composable.
I Proof. Let pAi WA
i=1 and pBj WB
j=1 two sets of probabilities associatedwith two statistically independent systems A and B. The jointprobability is given by
pA∪Bij = pAi · pBjThe total number of states of the composed system (including stateswith possibly null probability), is WAB = WAWB . We have
SU(A ∪ B) := kB
WA∑i=1
WB∑j=1
pA∪Bij G
(ln
1
pA∪Bij
)=
= kB
WA∑i=1
WB∑j=1
pAi ·pBj G
(ln
1
pAi+ ln
1
pBj
)= kB
WA∑i=1
WB∑j=1
pAi ·pBj G (t1 + t2) =
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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= kB
WA∑i=1
WB∑j=1
pAi · pB
j G(
G−1(s1) + G−1(s2))
= kB
WA∑i=1
WB∑j=1
pAi · pB
j Φ (s1, s2) =
= kB
WA∑i=1
WB∑j=1
pAi · pB
j
s1 + s2 +∞∑
k,m=1
ckmsk1 sm2
=
= kB
WA∑i=1
WB∑j=1
pAi · pB
j
[G
(ln
1
pAi
)+ G
(ln
1
pBj
)+ c11G
(ln
1
pAi
)· G
(ln
1
pBj
)+ . . .
]
The Boltzmann entropy corresponds to the case (i.e. ckm = 0 ∀k,m). TheTsallis entropy corresponds to the case c11 6= 0 and ckm = 0 for(k,m) 6= (1, 1). In both cases, we get immediately
SU(A + B) = Φ (SU(A), SU(B)) , (8)
with Φ(x , y) = x + y for ckm = 0, and Φ(x , y) = x + y + c11xy for c11 6= 0 andckm = 0. However, whenever ckm 6= 0 for (k,m) 6= (1, 1), a priori formula (8)does not hold in general (i.e. for any possible choice of pA
i WAi=1 and pB
j WBj=1).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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This because
Yk
(pA1 , . . . , p
Aw
)=
WA∑i=1
pAi Gk
(ln
1
pAi
)−
[WA∑i=1
pAi G
(ln
1
pAi
)]k6= 0!!.
Nevertheless, in the case that pAi and pBj are both the uniformdistribution, formula (8) holds for the whole family of entropiesrepresented by (6).
The formal power series φ(x , y) = G (G−1(x) + G−1(y)) for any choiceof G defines a formal group law. It verifies automatically the conditionsof symmetry, null composability and transitivity. We deduce that SU isweakly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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The universal-group entropyMain properties of SU entropyRelation with other entropies
Theorem 2.5
I The entropy SU [p] satisfies the Shannon-Khinchin axioms(SK1)-(SK3).
I (SK1). The group exponential G(t) is a real analytic function of t.Consequently, the universal group entropy is (at least) a continuousfunction of its arguments (p1, . . . , pw ).
I (SK2). The entropy SU [p] is supposed to be concave by definition.Nevertheless, we wish to prove here that condition (19), which allows toconstruct a large subclass of entropies of the form (6), is sufficient toguarantee concavity. To this aim, consider the quantity∑∞
k=0akk+1
x(ln 1
x
)k+1. By imposing strict concavity we get
− 1
xa0−a1+(a1−2a2) ln
1
x+(a2−3a3)
(ln
1
x
)2
+(a3−4a4)
(ln
1
x
)3
+. . . < 0.
(9)Condition ak > (k + 1)ak+1 ensures that the inequality (55) is satisfied.Consequently, the associated entropy (6) is strictly concave in its space ofparameters. Other choices of the sequence akk∈N are clearly possible.
I (SK3). Since by construction G(0) = 0, and limx→0 x(ln 1
x
)k= 0, it
follows that SU(0) = 0. Similarly, SU(1) = 0.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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The universal-group entropyMain properties of SU entropyRelation with other entropies
Theorem 2.5
I The entropy SU [p] satisfies the Shannon-Khinchin axioms(SK1)-(SK3).
I (SK1). The group exponential G(t) is a real analytic function of t.Consequently, the universal group entropy is (at least) a continuousfunction of its arguments (p1, . . . , pw ).
I (SK2). The entropy SU [p] is supposed to be concave by definition.Nevertheless, we wish to prove here that condition (19), which allows toconstruct a large subclass of entropies of the form (6), is sufficient toguarantee concavity. To this aim, consider the quantity∑∞
k=0akk+1
x(ln 1
x
)k+1. By imposing strict concavity we get
− 1
xa0−a1+(a1−2a2) ln
1
x+(a2−3a3)
(ln
1
x
)2
+(a3−4a4)
(ln
1
x
)3
+. . . < 0.
(9)Condition ak > (k + 1)ak+1 ensures that the inequality (55) is satisfied.Consequently, the associated entropy (6) is strictly concave in its space ofparameters. Other choices of the sequence akk∈N are clearly possible.
I (SK3). Since by construction G(0) = 0, and limx→0 x(ln 1
x
)k= 0, it
follows that SU(0) = 0. Similarly, SU(1) = 0.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Theorem 2.5
I The entropy SU [p] satisfies the Shannon-Khinchin axioms(SK1)-(SK3).
I (SK1). The group exponential G(t) is a real analytic function of t.Consequently, the universal group entropy is (at least) a continuousfunction of its arguments (p1, . . . , pw ).
I (SK2). The entropy SU [p] is supposed to be concave by definition.Nevertheless, we wish to prove here that condition (19), which allows toconstruct a large subclass of entropies of the form (6), is sufficient toguarantee concavity. To this aim, consider the quantity∑∞
k=0akk+1
x(ln 1
x
)k+1. By imposing strict concavity we get
− 1
xa0−a1+(a1−2a2) ln
1
x+(a2−3a3)
(ln
1
x
)2
+(a3−4a4)
(ln
1
x
)3
+. . . < 0.
(9)Condition ak > (k + 1)ak+1 ensures that the inequality (55) is satisfied.Consequently, the associated entropy (6) is strictly concave in its space ofparameters. Other choices of the sequence akk∈N are clearly possible.
I (SK3). Since by construction G(0) = 0, and limx→0 x(ln 1
x
)k= 0, it
follows that SU(0) = 0. Similarly, SU(1) = 0.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Theorem 2.5
I The entropy SU [p] satisfies the Shannon-Khinchin axioms(SK1)-(SK3).
I (SK1). The group exponential G(t) is a real analytic function of t.Consequently, the universal group entropy is (at least) a continuousfunction of its arguments (p1, . . . , pw ).
I (SK2). The entropy SU [p] is supposed to be concave by definition.Nevertheless, we wish to prove here that condition (19), which allows toconstruct a large subclass of entropies of the form (6), is sufficient toguarantee concavity. To this aim, consider the quantity∑∞
k=0akk+1
x(ln 1
x
)k+1. By imposing strict concavity we get
− 1
xa0−a1+(a1−2a2) ln
1
x+(a2−3a3)
(ln
1
x
)2
+(a3−4a4)
(ln
1
x
)3
+. . . < 0.
(9)Condition ak > (k + 1)ak+1 ensures that the inequality (55) is satisfied.Consequently, the associated entropy (6) is strictly concave in its space ofparameters. Other choices of the sequence akk∈N are clearly possible.
I (SK3). Since by construction G(0) = 0, and limx→0 x(ln 1
x
)k= 0, it
follows that SU(0) = 0. Similarly, SU(1) = 0.Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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Theorem 2.6 The entropy SU [p] is extensive: on the uniformprobability distribution (i.e. pi = 1/W for all i = 1, . . . ,W ), we have
SU [W ] = kBG (lnW ) ∼ N ⇐⇒W (N) ∼ exp(G−1(N)
). (10)
We should also require that W (N) ∼ exp(G−1(N)
), to guarantee that
W (N) is interpretable as an occupation law.Viceversa, given an occupation law W (N), we get G−1(x) = lnW (x);the inverse of this expression gives us G (x). In this way, we can designan entropy extensive in a given regime.These properties are all necessary for the applicability of the entropy (6)in thermodynamical contexts.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Relation with other entropies
I General approach. An useful property of the entropy SU [p] is thatit admits the following decomposition
SU [p] =∞∑k=1
γkSk [p], γk = ak−1/k, (11)
in terms of a set of elementary functionals
Sk [p] := kB∑W
i=1 pi(
ln 1pi
)k, with k ∈ N.
I Therefore we have the general expression
SU [p] = kB
W∑i=1
pi
∞∑k=1
ak−1k
(ln
1
pi
)k
I This analytic form is shared by essentially all the entropies known inthe literature, and allows a quick comparison among them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Relation with other entropies
I General approach. An useful property of the entropy SU [p] is thatit admits the following decomposition
SU [p] =∞∑k=1
γkSk [p], γk = ak−1/k, (11)
in terms of a set of elementary functionals
Sk [p] := kB∑W
i=1 pi(
ln 1pi
)k, with k ∈ N.
I Therefore we have the general expression
SU [p] = kB
W∑i=1
pi
∞∑k=1
ak−1k
(ln
1
pi
)k
I This analytic form is shared by essentially all the entropies known inthe literature, and allows a quick comparison among them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Relation with other entropies
I General approach. An useful property of the entropy SU [p] is thatit admits the following decomposition
SU [p] =∞∑k=1
γkSk [p], γk = ak−1/k, (11)
in terms of a set of elementary functionals
Sk [p] := kB∑W
i=1 pi(
ln 1pi
)k, with k ∈ N.
I Therefore we have the general expression
SU [p] = kB
W∑i=1
pi
∞∑k=1
ak−1k
(ln
1
pi
)k
I This analytic form is shared by essentially all the entropies known inthe literature, and allows a quick comparison among them.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The Boltzmann-Gibbs entropy
I a) The Boltzmann-Gibbs entropy
SB[p] = kB
W∑i=1
pi ln1
pi. (12)
It has the general form∑
i piG(
ln 1pi
), with
G (t) = t
(i.e. for a1 = 1, ai = 0, ∀i = 2, 3, . . .). The delta operatorassociated with G (t) according to Rota’s isomorphism is t = ∂
I The group structure is simply the additive group law
Φ(x , y) = G (G−1(x) + G−1(y)) = x + y .
Therefore:
S(A ∪ B) = Φ(S(A),S(B)) = S(A) + S(B).
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The Boltzmann-Gibbs entropy
I a) The Boltzmann-Gibbs entropy
SB[p] = kB
W∑i=1
pi ln1
pi. (12)
It has the general form∑
i piG(
ln 1pi
), with
G (t) = t
(i.e. for a1 = 1, ai = 0, ∀i = 2, 3, . . .). The delta operatorassociated with G (t) according to Rota’s isomorphism is t = ∂
I The group structure is simply the additive group law
Φ(x , y) = G (G−1(x) + G−1(y)) = x + y .
Therefore:
S(A ∪ B) = Φ(S(A),S(B)) = S(A) + S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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The group-theoretical approach for the Tsallis entropy
C. Tsallis, J. Stat. Phys. 52, 479-487 (1988).I b) The Tsallis entropy
Sq[p] = kB
∑Wi=1 p
qi − 1
1− q=
W∑i=1
pi lnq1
pi, 0 ≤ q < 1.
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp[(1− q)t]− 1
1− q= t +
1
2(1− q)t2 +
1
6(1− q)2t3 + . . .
I Group logarithm: lnq x = x1−q−11−q .
I The associated finite difference delta operator is δ+ = Tσ−1σ , with
σ = 1− q.I When q → 1, G (t)→ t and Tsallis’ entropy reduces to Boltzmann’s
entropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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On the asymptotic behaviour of generalized entropiesState of Art
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The group-theoretical approach for the Tsallis entropy
C. Tsallis, J. Stat. Phys. 52, 479-487 (1988).I b) The Tsallis entropy
Sq[p] = kB
∑Wi=1 p
qi − 1
1− q=
W∑i=1
pi lnq1
pi, 0 ≤ q < 1.
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp[(1− q)t]− 1
1− q= t +
1
2(1− q)t2 +
1
6(1− q)2t3 + . . .
I Group logarithm: lnq x = x1−q−11−q .
I The associated finite difference delta operator is δ+ = Tσ−1σ , with
σ = 1− q.I When q → 1, G (t)→ t and Tsallis’ entropy reduces to Boltzmann’s
entropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The group-theoretical approach for the Tsallis entropy
C. Tsallis, J. Stat. Phys. 52, 479-487 (1988).I b) The Tsallis entropy
Sq[p] = kB
∑Wi=1 p
qi − 1
1− q=
W∑i=1
pi lnq1
pi, 0 ≤ q < 1.
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp[(1− q)t]− 1
1− q= t +
1
2(1− q)t2 +
1
6(1− q)2t3 + . . .
I Group logarithm: lnq x = x1−q−11−q .
I The associated finite difference delta operator is δ+ = Tσ−1σ , with
σ = 1− q.I When q → 1, G (t)→ t and Tsallis’ entropy reduces to Boltzmann’s
entropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The group-theoretical approach for the Tsallis entropy
C. Tsallis, J. Stat. Phys. 52, 479-487 (1988).I b) The Tsallis entropy
Sq[p] = kB
∑Wi=1 p
qi − 1
1− q=
W∑i=1
pi lnq1
pi, 0 ≤ q < 1.
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp[(1− q)t]− 1
1− q= t +
1
2(1− q)t2 +
1
6(1− q)2t3 + . . .
I Group logarithm: lnq x = x1−q−11−q .
I The associated finite difference delta operator is δ+ = Tσ−1σ , with
σ = 1− q.
I When q → 1, G (t)→ t and Tsallis’ entropy reduces to Boltzmann’sentropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The group-theoretical approach for the Tsallis entropy
C. Tsallis, J. Stat. Phys. 52, 479-487 (1988).I b) The Tsallis entropy
Sq[p] = kB
∑Wi=1 p
qi − 1
1− q=
W∑i=1
pi lnq1
pi, 0 ≤ q < 1.
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp[(1− q)t]− 1
1− q= t +
1
2(1− q)t2 +
1
6(1− q)2t3 + . . .
I Group logarithm: lnq x = x1−q−11−q .
I The associated finite difference delta operator is δ+ = Tσ−1σ , with
σ = 1− q.I When q → 1, G (t)→ t and Tsallis’ entropy reduces to Boltzmann’s
entropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I Its decomposition is provided by:
Sq[p] = kB
W∑i=1
piln1
pi+
1
2(1− q)
(ln
1
pi
)2
+1
6(1− q)2
(ln
1
pi
)3
+ . . .
I Let us determine the associated group law:
G−1(s) =1
1− qln(1 + (1− q)s)
G−1(x) + G−1(y) =1
1− qln(
1 + (1− q)x + (1− q)y + (1− q)2xy)
Consequently we get
Φ(x , y) = G(G−1(x) + G−1(y)) = x + y + (1− q)xy .
which is nothing but the multiplicative group law. Once again, wededuce
S(A ∪ B) = Φ(S(A),S(B)) = S(A) + S(B) + (1− q)S(A)S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I Its decomposition is provided by:
Sq[p] = kB
W∑i=1
piln1
pi+
1
2(1− q)
(ln
1
pi
)2
+1
6(1− q)2
(ln
1
pi
)3
+ . . .
I Let us determine the associated group law:
G−1(s) =1
1− qln(1 + (1− q)s)
G−1(x) + G−1(y) =1
1− qln(
1 + (1− q)x + (1− q)y + (1− q)2xy)
Consequently we get
Φ(x , y) = G(G−1(x) + G−1(y)) = x + y + (1− q)xy .
which is nothing but the multiplicative group law. Once again, wededuce
S(A ∪ B) = Φ(S(A), S(B)) = S(A) + S(B) + (1− q)S(A)S(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The Kaniadakis entropy
G. Kaniadakis, Phys. Rev. E, 66, 056125 (2002).I c) The Kaniadakis entropy
Sκ[p] = kB
W∑i=1
pip−κi − pκi
2κ. (13)
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp(κt)− exp(−κt)
2κ
I The group logarithm is lnk x = xk−x−k
2k . The associated finite
difference operator is δ+ = Tσ−T−σ
2σ . Its decomposition is
Sκ[p] = kB
W∑i=1
pi
ln
1
pi+
1
3!κ2(
ln1
pi
)3
+1
5!κ4(
ln1
pi
)5
+ . . .
(14)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The Kaniadakis entropy
G. Kaniadakis, Phys. Rev. E, 66, 056125 (2002).I c) The Kaniadakis entropy
Sκ[p] = kB
W∑i=1
pip−κi − pκi
2κ. (13)
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp(κt)− exp(−κt)
2κ
I The group logarithm is lnk x = xk−x−k
2k . The associated finite
difference operator is δ+ = Tσ−T−σ
2σ . Its decomposition is
Sκ[p] = kB
W∑i=1
pi
ln
1
pi+
1
3!κ2(
ln1
pi
)3
+1
5!κ4(
ln1
pi
)5
+ . . .
(14)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The Kaniadakis entropy
G. Kaniadakis, Phys. Rev. E, 66, 056125 (2002).I c) The Kaniadakis entropy
Sκ[p] = kB
W∑i=1
pip−κi − pκi
2κ. (13)
I It has the general form∑
i piG(
ln 1pi
), with
G (t) =exp(κt)− exp(−κt)
2κ
I The group logarithm is lnk x = xk−x−k
2k . The associated finite
difference operator is δ+ = Tσ−T−σ
2σ . Its decomposition is
Sκ[p] = kB
W∑i=1
pi
ln
1
pi+
1
3!κ2(
ln1
pi
)3
+1
5!κ4(
ln1
pi
)5
+ . . .
(14)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Weak composability of Sk
Nota Bene. Kaniadakis’ entropy is not composable in full generality.
I Indeed,Sk(A ∪ B) 6= F (Sk(A),Sk(B), κ)
for any function F (x , y), if we wish that the previous property holdsfor any possible probability distribution piWi=1.
I However, if we choose the uniform distribution, then
Sk [p] = kB lnk W ,
andSk(Au ∪ Bu) = Φ(Sk(Au),Sk(Bu))
I whereΦ(x , y) = x
√1 + k2y2 + y
√1 + k2x2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Weak composability of Sk
Nota Bene. Kaniadakis’ entropy is not composable in full generality.
I Indeed,Sk(A ∪ B) 6= F (Sk(A),Sk(B), κ)
for any function F (x , y), if we wish that the previous property holdsfor any possible probability distribution piWi=1.
I However, if we choose the uniform distribution, then
Sk [p] = kB lnk W ,
andSk(Au ∪ Bu) = Φ(Sk(Au),Sk(Bu))
I whereΦ(x , y) = x
√1 + k2y2 + y
√1 + k2x2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
Weak composability of Sk
Nota Bene. Kaniadakis’ entropy is not composable in full generality.
I Indeed,Sk(A ∪ B) 6= F (Sk(A),Sk(B), κ)
for any function F (x , y), if we wish that the previous property holdsfor any possible probability distribution piWi=1.
I However, if we choose the uniform distribution, then
Sk [p] = kB lnk W ,
andSk(Au ∪ Bu) = Φ(Sk(Au),Sk(Bu))
I whereΦ(x , y) = x
√1 + k2y2 + y
√1 + k2x2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The S(c , d) entropy
R. Hanel and S. Thurner, Europhys. Lett. 93 20006 (2011).
I d) The Sc,d entropy reads
Sc,d =e
1− c + cd
W∑i=1
Γ(1+d , 1−c ln pi )−c
1− c + cd, c ∈ (0, 1], d ∈ R.
(15)
I We remind that the upper incomplete gamma function Γ(s, x) and theupper incomplete gamma functions are, respectively :
Γ(s, x) =
∫ ∞x
ts−1e−tdt, γ(s, x) =
∫ x
0
ts−1e−tdt, s ∈ C, Re s > 0.
I To study the group-theoretical structure of the Sc,d entropy, observe that
Γ(1 + d , 1− c ln pi ) = Γ(1 + d) + td∞∑k=0
δktk+1
k + 1, (16)
δk =(−1)k+1(k + 1)
k!(k + d + 1), t =
[ln
e
pic
]. (17)
In the subsequent considerations, we shall restrict to the case d ∈ N.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The S(c , d) entropy
R. Hanel and S. Thurner, Europhys. Lett. 93 20006 (2011).
I d) The Sc,d entropy reads
Sc,d =e
1− c + cd
W∑i=1
Γ(1+d , 1−c ln pi )−c
1− c + cd, c ∈ (0, 1], d ∈ R.
(15)
I We remind that the upper incomplete gamma function Γ(s, x) and theupper incomplete gamma functions are, respectively :
Γ(s, x) =
∫ ∞x
ts−1e−tdt, γ(s, x) =
∫ x
0
ts−1e−tdt, s ∈ C, Re s > 0.
I To study the group-theoretical structure of the Sc,d entropy, observe that
Γ(1 + d , 1− c ln pi ) = Γ(1 + d) + td∞∑k=0
δktk+1
k + 1, (16)
δk =(−1)k+1(k + 1)
k!(k + d + 1), t =
[ln
e
pic
]. (17)
In the subsequent considerations, we shall restrict to the case d ∈ N.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The S(c , d) entropy
R. Hanel and S. Thurner, Europhys. Lett. 93 20006 (2011).
I d) The Sc,d entropy reads
Sc,d =e
1− c + cd
W∑i=1
Γ(1+d , 1−c ln pi )−c
1− c + cd, c ∈ (0, 1], d ∈ R.
(15)
I We remind that the upper incomplete gamma function Γ(s, x) and theupper incomplete gamma functions are, respectively :
Γ(s, x) =
∫ ∞x
ts−1e−tdt, γ(s, x) =
∫ x
0
ts−1e−tdt, s ∈ C, Re s > 0.
I To study the group-theoretical structure of the Sc,d entropy, observe that
Γ(1 + d , 1− c ln pi ) = Γ(1 + d) + td∞∑k=0
δktk+1
k + 1, (16)
δk =(−1)k+1(k + 1)
k!(k + d + 1), t =
[ln
e
pic
]. (17)
In the subsequent considerations, we shall restrict to the case d ∈ N.Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I To perform our analysis we shall use the identity, valid for d ∈ N:∫ ∞K
tde−tdt = e−Kd∑
n=0
∏nj=0(d − j + 1)
d + 1K d−n. (18)
This identity can be proven by a direct computation. It allows to expandthe entropy in terms of the set Sk [p].
I In general, we have
Sc,d =1
1− c + cd
W∑i=1
pic
d∑
k=0
1
d + 1
k∏j=0
(d − j + 1)d−k∑n=0
(d − k
n
)(ln
1
pic
)n
− c
1− c + cd, d ∈ N.
I Particular cases of the previous formula:
S1,1[p] = 1+∑i
pi ln1
pi, S1,2[p] = 2
(1 +
∑i
pi ln1
pi
)+
1
2
∑i
pi
(ln
1
pi
)2
.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I To perform our analysis we shall use the identity, valid for d ∈ N:∫ ∞K
tde−tdt = e−Kd∑
n=0
∏nj=0(d − j + 1)
d + 1K d−n. (18)
This identity can be proven by a direct computation. It allows to expandthe entropy in terms of the set Sk [p].
I In general, we have
Sc,d =1
1− c + cd
W∑i=1
pic
d∑
k=0
1
d + 1
k∏j=0
(d − j + 1)d−k∑n=0
(d − k
n
)(ln
1
pic
)n
− c
1− c + cd, d ∈ N.
I Particular cases of the previous formula:
S1,1[p] = 1+∑i
pi ln1
pi, S1,2[p] = 2
(1 +
∑i
pi ln1
pi
)+
1
2
∑i
pi
(ln
1
pi
)2
.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I To perform our analysis we shall use the identity, valid for d ∈ N:∫ ∞K
tde−tdt = e−Kd∑
n=0
∏nj=0(d − j + 1)
d + 1K d−n. (18)
This identity can be proven by a direct computation. It allows to expandthe entropy in terms of the set Sk [p].
I In general, we have
Sc,d =1
1− c + cd
W∑i=1
pic
d∑
k=0
1
d + 1
k∏j=0
(d − j + 1)d−k∑n=0
(d − k
n
)(ln
1
pic
)n
− c
1− c + cd, d ∈ N.
I Particular cases of the previous formula:
S1,1[p] = 1+∑i
pi ln1
pi, S1,2[p] = 2
(1 +
∑i
pi ln1
pi
)+
1
2
∑i
pi
(ln
1
pi
)2
.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I For c arbitrary, let us write explicitly, for instance, the functionalscorresponding to d = 3:
Sc,3[p] =1
1 + 2c
W∑i=1
pic
16 + 15 ln
1
pic
+ 6
(ln
1
pic
)2
+
(ln
1
pic
)3− c
1 + 2c,
We have:G(t) = 16 + 15t + 6t2 + t3
I and d = 5:
Sc,5[p] =1
1 + 4c
W∑i=1
pic
326 + 325 ln
1
pic
+ 160
(ln
1
pic
)2
+ 50
(ln
1
pic
)3
+ 10
(ln
1
pic
)4
+
(ln
1
pic
)5− c
1 + 4c.
withG(t) = 326 + 325t + 160t2 + 50t3 + 10t4 + t5
I In both cases:
ak > (k + 1)ak+1 ∀k ∈ N with akk∈N ≥ 0 (19)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I For c arbitrary, let us write explicitly, for instance, the functionalscorresponding to d = 3:
Sc,3[p] =1
1 + 2c
W∑i=1
pic
16 + 15 ln
1
pic
+ 6
(ln
1
pic
)2
+
(ln
1
pic
)3− c
1 + 2c,
We have:G(t) = 16 + 15t + 6t2 + t3
I and d = 5:
Sc,5[p] =1
1 + 4c
W∑i=1
pic
326 + 325 ln
1
pic
+ 160
(ln
1
pic
)2
+ 50
(ln
1
pic
)3
+ 10
(ln
1
pic
)4
+
(ln
1
pic
)5− c
1 + 4c.
withG(t) = 326 + 325t + 160t2 + 50t3 + 10t4 + t5
I In both cases:
ak > (k + 1)ak+1 ∀k ∈ N with akk∈N ≥ 0 (19)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I For c arbitrary, let us write explicitly, for instance, the functionalscorresponding to d = 3:
Sc,3[p] =1
1 + 2c
W∑i=1
pic
16 + 15 ln
1
pic
+ 6
(ln
1
pic
)2
+
(ln
1
pic
)3− c
1 + 2c,
We have:G(t) = 16 + 15t + 6t2 + t3
I and d = 5:
Sc,5[p] =1
1 + 4c
W∑i=1
pic
326 + 325 ln
1
pic
+ 160
(ln
1
pic
)2
+ 50
(ln
1
pic
)3
+ 10
(ln
1
pic
)4
+
(ln
1
pic
)5− c
1 + 4c.
withG(t) = 326 + 325t + 160t2 + 50t3 + 10t4 + t5
I In both cases:
ak > (k + 1)ak+1 ∀k ∈ N with akk∈N ≥ 0 (19)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
General group properties
From the previous analysis, it emerges that the S(c , d) entropy fits intothe class SU in the two cases
I a) (c = 1, d ∈ N) (we can get rid of the constant term in theexpansion)
I b) (c > 0, d = 0).
I In the first case, we get that the S(c , d) is weakly composable.The group structure Φ(x , y) is generated by functions G (t) that arepolynomials. The explicit expression of G (t) can be explicitlycomputed, but it is usually cumbersome.
I For other cases, S(c , d) does not seem to be weakly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
General group properties
From the previous analysis, it emerges that the S(c , d) entropy fits intothe class SU in the two cases
I a) (c = 1, d ∈ N) (we can get rid of the constant term in theexpansion)
I b) (c > 0, d = 0).
I In the first case, we get that the S(c , d) is weakly composable.The group structure Φ(x , y) is generated by functions G (t) that arepolynomials. The explicit expression of G (t) can be explicitlycomputed, but it is usually cumbersome.
I For other cases, S(c , d) does not seem to be weakly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
General group properties
From the previous analysis, it emerges that the S(c , d) entropy fits intothe class SU in the two cases
I a) (c = 1, d ∈ N) (we can get rid of the constant term in theexpansion)
I b) (c > 0, d = 0).
I In the first case, we get that the S(c , d) is weakly composable.The group structure Φ(x , y) is generated by functions G (t) that arepolynomials. The explicit expression of G (t) can be explicitlycomputed, but it is usually cumbersome.
I For other cases, S(c , d) does not seem to be weakly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
General group properties
From the previous analysis, it emerges that the S(c , d) entropy fits intothe class SU in the two cases
I a) (c = 1, d ∈ N) (we can get rid of the constant term in theexpansion)
I b) (c > 0, d = 0).
I In the first case, we get that the S(c , d) is weakly composable.The group structure Φ(x , y) is generated by functions G (t) that arepolynomials. The explicit expression of G (t) can be explicitlycomputed, but it is usually cumbersome.
I For other cases, S(c , d) does not seem to be weakly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
E. P. Borges and I. Roditi, Phys. Lett. A 246 399–402 (1998).
I e) The Borges-Roditi entropy is a two-parametric entropy. Theassociated generalized logarithm reads
Loga,b(x) =xa − xb
a− b, (20)
which reproduces Abe’s entropy for a = σ − 1, b = σ−1 − 1. The related
group exponential is GA(t) = eat−ebt
a−b. The associated formal group is
known in the literature as the Abel formal group:
ΦA(x , y) = x + y + β1xy +∑j>i
βi(
xy i − x iy). (21)
The coefficients βn in (21) can be expressed as
βn =(−1)n−1
n!(n − 1)
∏i+j=n−1, i,j≥0
(ia + jb).
SBR [p] = kB
W∑i=1
pi
ln
1
pi+
1
2(a + b)
(ln
1
pi
)2
+1
6
(a2 + ab + b2
)(ln
1
pi
)3
. . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
The Tsallis-Cirto entropy
C. Tsallis, Introduction to nonextensive statistical mechanics, Springer, 2009 ;C. Tsallis, L. Cirto, Eur. Phys. J. C 73, 2487 (2013); M. R. Ubriaco, Phys.Lett. A 373, 2516–2519 2009.f) The Sδ entropy reads
Sδ = kB
W∑i=1
pi
(ln
1
pi
)δ, 0 < δ ≤ (1 + ln W ). (22)
The underlying algebraic structure can be analyzed on the uniform distribution:Sδ is weakly composable. We get The case δ = 1 is the only strictlycomposable case. Also, Φ(x , y) does not define a group law over the reals,but simply a monoid, except for δ ∈ N, δ odd. A similar analysis can beperformed for the case of the entropic functional
Sq,δ = kB
W∑i=1
pi
(lnq
1
pi
)δ. (23)
It reduces to the Tsallis entropy for q ∈ R and δ = 1.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
g) Entropies from delta operators
As we have seen, from a class of discrete derivatives we can define, via theRota isomorphism, the generalized logarithm
LogG (x) =1
σ
m∑n=l
knxσn, l ,m ∈ Z, m − l = r > 0, x > 0. (24)
We can associate with these logarithms the class of entropies
SG (p) := kB
W∑i=1
piLogG
(1
pi
). (25)
where kn are real constants such thatm∑n=l
kn = 0,m∑n=l
nkn = 1, (26)
and km 6= 0, kl 6= 0. Conditions (26) ensure that limσ→0 LogG (x) = ln x .The class (25) is a representation of SU entropy, corresponding to the choice
G(t) =1
σ
m∑n=l
kn exp(nσt), l , m ∈ Z, m − l = r > 0, (27)
with the constraints (26).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
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On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I Let us construct some new examples. Consider the delta operators
∆III = T−2T−1+T−2
σ, ∆IV =
T 2− 32T+ 3
2T−1−T−2
σ,
∆V = T 3−2T 2+2T−2T−1+T−2
−σ , . . .
I The corresponding logarithms are
LogGIII (x) =1
σ
(xσ − 2x−σ + x−2σ
),
LogGIV (x) =1
σ
(x2σ − 3
2xσ +
3
2x−σ − x−2σ
),
LogGV (x) =1
σ
(x3σ − 2x2σ + 2xσ − 2x−σ + x−2σ
).
I Consequently, we introduce the new entropic forms
SGIII (p) :=k
σ
N∑i=1
pi
(p2σi − 2pσi + p−σi
), (28)
SGIV (p) :=k
σ
N∑i=1
pi
(p−2σi − 3
2p−σi +
3
2pσi − p2σ
i
), (29)
SGV (p) :=k
σ
N∑i=1
pi
(p−3σi − 2p−2σ
i + 2p−σi − 2pσi + p2σi
), . . .
(30)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I Let us construct some new examples. Consider the delta operators
∆III = T−2T−1+T−2
σ, ∆IV =
T 2− 32T+ 3
2T−1−T−2
σ,
∆V = T 3−2T 2+2T−2T−1+T−2
−σ , . . .
I The corresponding logarithms are
LogGIII (x) =1
σ
(xσ − 2x−σ + x−2σ
),
LogGIV (x) =1
σ
(x2σ − 3
2xσ +
3
2x−σ − x−2σ
),
LogGV (x) =1
σ
(x3σ − 2x2σ + 2xσ − 2x−σ + x−2σ
).
I Consequently, we introduce the new entropic forms
SGIII (p) :=k
σ
N∑i=1
pi
(p2σi − 2pσi + p−σi
), (28)
SGIV (p) :=k
σ
N∑i=1
pi
(p−2σi − 3
2p−σi +
3
2pσi − p2σ
i
), (29)
SGV (p) :=k
σ
N∑i=1
pi
(p−3σi − 2p−2σ
i + 2p−σi − 2pσi + p2σi
), . . .
(30)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The universal-group entropyMain properties of SU entropyRelation with other entropies
I Let us construct some new examples. Consider the delta operators
∆III = T−2T−1+T−2
σ, ∆IV =
T 2− 32T+ 3
2T−1−T−2
σ,
∆V = T 3−2T 2+2T−2T−1+T−2
−σ , . . .
I The corresponding logarithms are
LogGIII (x) =1
σ
(xσ − 2x−σ + x−2σ
),
LogGIV (x) =1
σ
(x2σ − 3
2xσ +
3
2x−σ − x−2σ
),
LogGV (x) =1
σ
(x3σ − 2x2σ + 2xσ − 2x−σ + x−2σ
).
I Consequently, we introduce the new entropic forms
SGIII (p) :=k
σ
N∑i=1
pi
(p2σi − 2pσi + p−σi
), (28)
SGIV (p) :=k
σ
N∑i=1
pi
(p−2σi − 3
2p−σi +
3
2pσi − p2σ
i
), (29)
SGV (p) :=k
σ
N∑i=1
pi
(p−3σi − 2p−2σ
i + 2p−σi − 2pσi + p2σi
), . . .
(30)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
Thermodynamics from the group structure
We shall discuss the maximization of group entropies under appropriateconstraints: we adopt a generalized maximum entropy principle. We willsee that the Legendre structure of classical thermodynamics ispreserved in the group-theoretical framework.Precisely, let
LogU [ε] = G (ln ε)
where G (t) is the power series (5), with the constraint (19). Consider anisolated system in a stationary state (microcanonical ensemble). Theoptimization of SU leads to the equal probability case, i.e. pi = 1/W ,∀i .Therefore, we have
SU [p] = kBLogUW , (31)
which reduces to the celebrated Boltzmann formula SBG = kB lnW inthe case of uncorrelated particles.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
On the Legendre structure
I Let us consider a system in thermal contact with a reservoir (canonicalensemble). We introduce the numbers εi , interpreted as the values of aphysically relevant observable, typically the value of the energy of thesystem in its ith state. Assume that pi (εi ) is a normalized andmonotonically decreasing distribution function of εi . The internal energyV in a given state is defined as V =
∑Wi=1 εipi (εi ).
I As usual, we shall study the variational problem of the existence of astationary distribution pi (ε). To this aim, we introduce the functional
L = SG [p]− α
[∑i
p(εi )− 1
]− β
[W∑i=1
εipi (εi )− V
], (32)
where α and β are Lagrange multipliers.
I The vanishing of the variational derivative of this functional with respectto the distribution pi provides the stationary solution
pi = γE
(−α− β(εi )
λ
), (33)
with γ a priori a normalization constant, and E(·) is an invertible function.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
On the Legendre structure
I Let us consider a system in thermal contact with a reservoir (canonicalensemble). We introduce the numbers εi , interpreted as the values of aphysically relevant observable, typically the value of the energy of thesystem in its ith state. Assume that pi (εi ) is a normalized andmonotonically decreasing distribution function of εi . The internal energyV in a given state is defined as V =
∑Wi=1 εipi (εi ).
I As usual, we shall study the variational problem of the existence of astationary distribution pi (ε). To this aim, we introduce the functional
L = SG [p]− α
[∑i
p(εi )− 1
]− β
[W∑i=1
εipi (εi )− V
], (32)
where α and β are Lagrange multipliers.
I The vanishing of the variational derivative of this functional with respectto the distribution pi provides the stationary solution
pi = γE
(−α− β(εi )
λ
), (33)
with γ a priori a normalization constant, and E(·) is an invertible function.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
On the Legendre structure
I Let us consider a system in thermal contact with a reservoir (canonicalensemble). We introduce the numbers εi , interpreted as the values of aphysically relevant observable, typically the value of the energy of thesystem in its ith state. Assume that pi (εi ) is a normalized andmonotonically decreasing distribution function of εi . The internal energyV in a given state is defined as V =
∑Wi=1 εipi (εi ).
I As usual, we shall study the variational problem of the existence of astationary distribution pi (ε). To this aim, we introduce the functional
L = SG [p]− α
[∑i
p(εi )− 1
]− β
[W∑i=1
εipi (εi )− V
], (32)
where α and β are Lagrange multipliers.
I The vanishing of the variational derivative of this functional with respectto the distribution pi provides the stationary solution
pi = γE
(−α− β(εi )
λ
), (33)
with γ a priori a normalization constant, and E(·) is an invertible function.Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I G. Kaniadakis, M. Lissia and A. M. Scarfone, Physica A 340 (2004)41–49 ; Phys. Rev. E 71 (2005) 046128 ; S. Abe and S. Thurner,Europhys. Lett 81 (2008) 1004 ; G. Kaniadakis, Eur. Phys. J. B 70 3–13(2009).
Let SG = −∑W
i=1 piLogG (pi ). The condition for identifying E with ageneralized logarithm is
d
dpj[pjLogG (pj)] = λLogG
(pj
γ
).
I For a specific class of entropies (Kaniadakis, Sharma-Mittal, etc.), onecan construct an ad-hoc Legendre structure (by using Legendre andMassieu potentials):
LogG (Z) + βVG = SG , Z =W∑i=1
E(−α− βεi )). (34)
I The previous equation can be used to introduce a thermodynamicobservable T , which has the interpretation of a local temperature for anon-equilibrium metastable state. Precisely, we can define ∂SU
∂V = 1T.
I Usually, a more accurate selection of the constraints is necessary (see C.Tsallis, R. Mendes, A. R. Plastino Physica A, 261 534-554 (1998)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I G. Kaniadakis, M. Lissia and A. M. Scarfone, Physica A 340 (2004)41–49 ; Phys. Rev. E 71 (2005) 046128 ; S. Abe and S. Thurner,Europhys. Lett 81 (2008) 1004 ; G. Kaniadakis, Eur. Phys. J. B 70 3–13(2009).
Let SG = −∑W
i=1 piLogG (pi ). The condition for identifying E with ageneralized logarithm is
d
dpj[pjLogG (pj)] = λLogG
(pj
γ
).
I For a specific class of entropies (Kaniadakis, Sharma-Mittal, etc.), onecan construct an ad-hoc Legendre structure (by using Legendre andMassieu potentials):
LogG (Z) + βVG = SG , Z =W∑i=1
E(−α− βεi )). (34)
I The previous equation can be used to introduce a thermodynamicobservable T , which has the interpretation of a local temperature for anon-equilibrium metastable state. Precisely, we can define ∂SU
∂V = 1T.
I Usually, a more accurate selection of the constraints is necessary (see C.Tsallis, R. Mendes, A. R. Plastino Physica A, 261 534-554 (1998)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I G. Kaniadakis, M. Lissia and A. M. Scarfone, Physica A 340 (2004)41–49 ; Phys. Rev. E 71 (2005) 046128 ; S. Abe and S. Thurner,Europhys. Lett 81 (2008) 1004 ; G. Kaniadakis, Eur. Phys. J. B 70 3–13(2009).
Let SG = −∑W
i=1 piLogG (pi ). The condition for identifying E with ageneralized logarithm is
d
dpj[pjLogG (pj)] = λLogG
(pj
γ
).
I For a specific class of entropies (Kaniadakis, Sharma-Mittal, etc.), onecan construct an ad-hoc Legendre structure (by using Legendre andMassieu potentials):
LogG (Z) + βVG = SG , Z =W∑i=1
E(−α− βεi )). (34)
I The previous equation can be used to introduce a thermodynamicobservable T , which has the interpretation of a local temperature for anon-equilibrium metastable state. Precisely, we can define ∂SU
∂V = 1T.
I Usually, a more accurate selection of the constraints is necessary (see C.Tsallis, R. Mendes, A. R. Plastino Physica A, 261 534-554 (1998)).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A new class of trace-form multidimensional entropies
How to design a trace-form entropy by using the notion ofuniversal-group entropy?
I I) Given a group law Φ(x , y) (a priori function of a set ofparameters), determine the function G such that the group law canbe represented in the form
Φ(x , y) = G (G−1(x) + G−1(y)).
I II) The corresponding entropy will be a specific representation ofSU [p], and is provided by
W∑i=1
piG
(ln
1
pi
)I III) Under mild hypotheses (concerning the set of parametersakn∈N), this entropy is weakly composable, satisfies the firstthree SK axioms, is extensive, possesses a Legendre structure,etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A new class of trace-form multidimensional entropies
How to design a trace-form entropy by using the notion ofuniversal-group entropy?
I I) Given a group law Φ(x , y) (a priori function of a set ofparameters), determine the function G such that the group law canbe represented in the form
Φ(x , y) = G (G−1(x) + G−1(y)).
I II) The corresponding entropy will be a specific representation ofSU [p], and is provided by
W∑i=1
piG
(ln
1
pi
)
I III) Under mild hypotheses (concerning the set of parametersakn∈N), this entropy is weakly composable, satisfies the firstthree SK axioms, is extensive, possesses a Legendre structure,etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A new class of trace-form multidimensional entropies
How to design a trace-form entropy by using the notion ofuniversal-group entropy?
I I) Given a group law Φ(x , y) (a priori function of a set ofparameters), determine the function G such that the group law canbe represented in the form
Φ(x , y) = G (G−1(x) + G−1(y)).
I II) The corresponding entropy will be a specific representation ofSU [p], and is provided by
W∑i=1
piG
(ln
1
pi
)I III) Under mild hypotheses (concerning the set of parametersakn∈N), this entropy is weakly composable, satisfies the firstthree SK axioms, is extensive, possesses a Legendre structure,etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A new entropy associated with the rational group law
E. Curado, P. Tempesta, C. Tsallis, arxiv: 1507.05058 (2015)
I There is a remarkable example of rational group law:
S(A ∪ B) =S(A) + S(B) + aS(A)S(B)
1 + bS(A)S(B), (35)
where a, b ∈ R.
I The corresponding formal group is given by
ΦR(x , y) =x + y + axy
1 + bxy(36)
When a = b = 0, we recover the additive law; for b = 0, we recover themultiplicative case. Whenever b 6= 0, we have a genuinely new case.
I A specific one-parametric realization of it is
Φ(x , y) =x + y + (α− 1)xy
1 + αxy(37)
For α = −1, 0, 1, we obtain group laws respectively associated with theEuler characteristic, the Todd genus and the Hirzebruch L-genus.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A new entropy associated with the rational group law
E. Curado, P. Tempesta, C. Tsallis, arxiv: 1507.05058 (2015)
I There is a remarkable example of rational group law:
S(A ∪ B) =S(A) + S(B) + aS(A)S(B)
1 + bS(A)S(B), (35)
where a, b ∈ R.
I The corresponding formal group is given by
ΦR(x , y) =x + y + axy
1 + bxy(36)
When a = b = 0, we recover the additive law; for b = 0, we recover themultiplicative case. Whenever b 6= 0, we have a genuinely new case.
I A specific one-parametric realization of it is
Φ(x , y) =x + y + (α− 1)xy
1 + αxy(37)
For α = −1, 0, 1, we obtain group laws respectively associated with theEuler characteristic, the Todd genus and the Hirzebruch L-genus.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
A new entropy associated with the rational group law
E. Curado, P. Tempesta, C. Tsallis, arxiv: 1507.05058 (2015)
I There is a remarkable example of rational group law:
S(A ∪ B) =S(A) + S(B) + aS(A)S(B)
1 + bS(A)S(B), (35)
where a, b ∈ R.
I The corresponding formal group is given by
ΦR(x , y) =x + y + axy
1 + bxy(36)
When a = b = 0, we recover the additive law; for b = 0, we recover themultiplicative case. Whenever b 6= 0, we have a genuinely new case.
I A specific one-parametric realization of it is
Φ(x , y) =x + y + (α− 1)xy
1 + αxy(37)
For α = −1, 0, 1, we obtain group laws respectively associated with theEuler characteristic, the Todd genus and the Hirzebruch L-genus.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I It is easy to verify that ΦR(x , y) satisfies the group axioms. Also, itadmits an inverse, i.e. there exists a real function φ(x) such that
ΦR(x , φ(x)) = 0. (38)
For instance, for the case of the additive law, φ(x) = −x . For the rationalgroup law(35), we have φ(x) = −x/(1 + ax)
I Problem: how to construct the entropic form associated with therational group law (35)?
I We look for a function G(t), which a priori is a formal power series, suchthat
ΦR(x , y) = G(G−1(x) + G−1(y)). (39)
I The most general form of G(t) is
G(t) =∞∑k=0
Aktk+1
k + 1= A0t + A1
t2
2+ A2
t3
3+ . . . . (40)
The inverse G−1(s) can be computed by means of the Lagrange inversiontheorem. We get:
G−1(s) =s
A0− A1
2(A0)3s2+ (41)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I It is easy to verify that ΦR(x , y) satisfies the group axioms. Also, itadmits an inverse, i.e. there exists a real function φ(x) such that
ΦR(x , φ(x)) = 0. (38)
For instance, for the case of the additive law, φ(x) = −x . For the rationalgroup law(35), we have φ(x) = −x/(1 + ax)
I Problem: how to construct the entropic form associated with therational group law (35)?
I We look for a function G(t), which a priori is a formal power series, suchthat
ΦR(x , y) = G(G−1(x) + G−1(y)). (39)
I The most general form of G(t) is
G(t) =∞∑k=0
Aktk+1
k + 1= A0t + A1
t2
2+ A2
t3
3+ . . . . (40)
The inverse G−1(s) can be computed by means of the Lagrange inversiontheorem. We get:
G−1(s) =s
A0− A1
2(A0)3s2+ (41)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I It is easy to verify that ΦR(x , y) satisfies the group axioms. Also, itadmits an inverse, i.e. there exists a real function φ(x) such that
ΦR(x , φ(x)) = 0. (38)
For instance, for the case of the additive law, φ(x) = −x . For the rationalgroup law(35), we have φ(x) = −x/(1 + ax)
I Problem: how to construct the entropic form associated with therational group law (35)?
I We look for a function G(t), which a priori is a formal power series, suchthat
ΦR(x , y) = G(G−1(x) + G−1(y)). (39)
I The most general form of G(t) is
G(t) =∞∑k=0
Aktk+1
k + 1= A0t + A1
t2
2+ A2
t3
3+ . . . . (40)
The inverse G−1(s) can be computed by means of the Lagrange inversiontheorem. We get:
G−1(s) =s
A0− A1
2(A0)3s2+ (41)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I It is easy to verify that ΦR(x , y) satisfies the group axioms. Also, itadmits an inverse, i.e. there exists a real function φ(x) such that
ΦR(x , φ(x)) = 0. (38)
For instance, for the case of the additive law, φ(x) = −x . For the rationalgroup law(35), we have φ(x) = −x/(1 + ax)
I Problem: how to construct the entropic form associated with therational group law (35)?
I We look for a function G(t), which a priori is a formal power series, suchthat
ΦR(x , y) = G(G−1(x) + G−1(y)). (39)
I The most general form of G(t) is
G(t) =∞∑k=0
Aktk+1
k + 1= A0t + A1
t2
2+ A2
t3
3+ . . . . (40)
The inverse G−1(s) can be computed by means of the Lagrange inversiontheorem. We get:
G−1(s) =s
A0− A1
2(A0)3s2+ (41)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
The expansion of the rational group law ΦR(x , y) is
ΦR(x , y) = x + y + axy − b(xy 2 + yx2)− abx2y 2
+ b2(x2y 3 + x3y 2) + ab2x3y 3 + higher order terms
Procedure. By computing the expression ΦR(x , y) = G(G−1(x) + G−1(y))
with the form G(t) = A0t + A1t2
2+ A2
t3
3+ . . . for the expansion of the formal
exponential, and identifying the terms appearing in this expansion with thosecoming from (42), we get an infinite set of relations for the coefficients Ak :
A0 ∈ RA1 = aA2
0
A2 =1
2
(a2 − 2b
)A3
0
A3 =1
3!
(a3 − 8ab
)A4
0
A4 =1
4!
(a4 − 22 a2b + 16 b2
)A5
0
A5 =1
5!
(a5 − 52 a3b + 136 ab2
)A6
0
... (42)
The coefficients Ak provide the most general solution to our problem.Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
Before proceeding further, let us consider the particular case b = 0. Then
Φ(x , y) = x + y + axy (43)
If we put b = 0, A0 = 1 in the previous coefficients (42), we getimmediately
ak =1
k!ak (44)
i.e.
G (t) =eat − 1
a, (45)
which is the correct form we were looking for! Indeed, using the
prescription S =∑W
i=1 piG(
ln 1pi
)and putting a = 1− q we get back the
Sq entropy
S =W∑i=1
pipq−1i − 1
1− q=
1−∑W
i=1 pqi
q − 1. (46)
The general case provides us with a series solution: indeed, wereconstruct G (t) and hence the entropy, term by term. However, thisseries can be re-summed up!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I We obtain the closed form solution
G(t) =2(ert − 1)
−a(ert − 1)±√
a2 + 4b(ert + 1)(47)
In particular, the coefficient A0 is fixed to be
A0 = ± r√a2 + 4b
. (48)
Consequently, we have deduced the following expression for the entropyassociated with ΦR(x , y).
I Definition 2.7. The S(+)a,b,r entropy, for r > 0, is the function
S(+)a,b,r [p] =
W∑i=1
s(+)[pi ] :=W∑i=1
piLog(+)a,b,r
(1
pi
), (49)
where the generalized plus logarithm is defined as
Log(+)a,b,r (x) :=
2(x r − 1)
−a(x r − 1) +√
a2 + 4b (x r + 1). (50)
A similar expression holds for the S(−)a,b,r entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I We obtain the closed form solution
G(t) =2(ert − 1)
−a(ert − 1)±√
a2 + 4b(ert + 1)(47)
In particular, the coefficient A0 is fixed to be
A0 = ± r√a2 + 4b
. (48)
Consequently, we have deduced the following expression for the entropyassociated with ΦR(x , y).
I Definition 2.7. The S(+)a,b,r entropy, for r > 0, is the function
S(+)a,b,r [p] =
W∑i=1
s(+)[pi ] :=W∑i=1
piLog(+)a,b,r
(1
pi
), (49)
where the generalized plus logarithm is defined as
Log(+)a,b,r (x) :=
2(x r − 1)
−a(x r − 1) +√
a2 + 4b (x r + 1). (50)
A similar expression holds for the S(−)a,b,r entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
Properties of the rational entropy S(+)a,b,r [p]
I Proposition. The entropies S(±)a,b,r reproduce the standard SBG entropy for
b = 0, a = r , in the limit r → 0:
limr→0
S(±)r,0,r [p] = SBG [p]. (51)
Proposition 2.8. The entropy S(+)a,b,r [p] satisfies the first three Khinchin
axioms.
I The extensivity problem
One of the main reasons to consider generalized entropies is the fact thatthey can be useful, or even mandatory, to describe systems with unusualbehavior. If an entropy is extensive, it essentially means that, for anoccupation law W = W (N) of the phase space associated with a givensystem, it is asymptotically proportional to N, the number of particles ofthe system. Precisely, SBG is extensive whenever W (N) ∼ kN , wherek ∈ R+ is a suitable constant. However, for substantially different choicesof W = W (N), this property is no longer true for SBG .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
Properties of the rational entropy S(+)a,b,r [p]
I Proposition. The entropies S(±)a,b,r reproduce the standard SBG entropy for
b = 0, a = r , in the limit r → 0:
limr→0
S(±)r,0,r [p] = SBG [p]. (51)
Proposition 2.8. The entropy S(+)a,b,r [p] satisfies the first three Khinchin
axioms.
I The extensivity problem
One of the main reasons to consider generalized entropies is the fact thatthey can be useful, or even mandatory, to describe systems with unusualbehavior. If an entropy is extensive, it essentially means that, for anoccupation law W = W (N) of the phase space associated with a givensystem, it is asymptotically proportional to N, the number of particles ofthe system. Precisely, SBG is extensive whenever W (N) ∼ kN , wherek ∈ R+ is a suitable constant. However, for substantially different choicesof W = W (N), this property is no longer true for SBG .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
The extensivity problem
I A natural question is to ascertain whether the new entropy we propose inthis paper is extensive. Its group structure, once again, ensures that thisproperty holds for a suitable asymptotic occupation law W = W (N) ofphase space. A general result of the group-theoretical approach is that a
sufficient condition for an entropy of the form S =∑W
i=1 piG(
ln 1pi
)to
be extensive is thatln W (N) ∼ G−1(N), (52)
provided that W (N) be real and defined for all N ∈ N, with W (N)→∞for N →∞. These requirements usually restrict the space of parameters.
I In our case, we observe that when pi = 1/W for all i = 1, · · · ,W , both
entropies S(±)a,b,r tend to the limit value
S(±)a,b,r [1/W ]→ 2√
a2 + 4b − a, (53)
if limN→∞W (N) =∞. In particular, for b → 0, the entropy diverges; asa consequence of the previous discussion, there exists a regime ofextensivity, for W (N) ∼ Nγ , with γ = 1/a.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
The extensivity problem
I A natural question is to ascertain whether the new entropy we propose inthis paper is extensive. Its group structure, once again, ensures that thisproperty holds for a suitable asymptotic occupation law W = W (N) ofphase space. A general result of the group-theoretical approach is that a
sufficient condition for an entropy of the form S =∑W
i=1 piG(
ln 1pi
)to
be extensive is thatln W (N) ∼ G−1(N), (52)
provided that W (N) be real and defined for all N ∈ N, with W (N)→∞for N →∞. These requirements usually restrict the space of parameters.
I In our case, we observe that when pi = 1/W for all i = 1, · · · ,W , both
entropies S(±)a,b,r tend to the limit value
S(±)a,b,r [1/W ]→ 2√
a2 + 4b − a, (53)
if limN→∞W (N) =∞. In particular, for b → 0, the entropy diverges; asa consequence of the previous discussion, there exists a regime ofextensivity, for W (N) ∼ Nγ , with γ = 1/a.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I If limN→∞W (N) = c ∈ R+, with c > 1, these entropies tend to the value
S(±)a,b,r [1/c]→ 2 (c r − 1)
±√
a2 + 4b (c r + 1)− a (c r − 1). (54)
I In both cases, if b 6= 0 the limiting value is finite, independently onwhether W diverges or tends to a constant for N →∞. Consequently, forb 6= 0 these entropies, albeit monotonically increasing functions of W , cannot be extensive.
I A natural question emerges, concerning the kind of systems such type ofentropies could be useful for.
I A possible answer is that the present formalism could be relevantwhenever treating systems highly correlated, where the addition of anew degree of freedom essentially does not change the value of theentropy, for a large number of degrees of freedom.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I If limN→∞W (N) = c ∈ R+, with c > 1, these entropies tend to the value
S(±)a,b,r [1/c]→ 2 (c r − 1)
±√
a2 + 4b (c r + 1)− a (c r − 1). (54)
I In both cases, if b 6= 0 the limiting value is finite, independently onwhether W diverges or tends to a constant for N →∞. Consequently, forb 6= 0 these entropies, albeit monotonically increasing functions of W , cannot be extensive.
I A natural question emerges, concerning the kind of systems such type ofentropies could be useful for.
I A possible answer is that the present formalism could be relevantwhenever treating systems highly correlated, where the addition of anew degree of freedom essentially does not change the value of theentropy, for a large number of degrees of freedom.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I If limN→∞W (N) = c ∈ R+, with c > 1, these entropies tend to the value
S(±)a,b,r [1/c]→ 2 (c r − 1)
±√
a2 + 4b (c r + 1)− a (c r − 1). (54)
I In both cases, if b 6= 0 the limiting value is finite, independently onwhether W diverges or tends to a constant for N →∞. Consequently, forb 6= 0 these entropies, albeit monotonically increasing functions of W , cannot be extensive.
I A natural question emerges, concerning the kind of systems such type ofentropies could be useful for.
I A possible answer is that the present formalism could be relevantwhenever treating systems highly correlated, where the addition of anew degree of freedom essentially does not change the value of theentropy, for a large number of degrees of freedom.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I If limN→∞W (N) = c ∈ R+, with c > 1, these entropies tend to the value
S(±)a,b,r [1/c]→ 2 (c r − 1)
±√
a2 + 4b (c r + 1)− a (c r − 1). (54)
I In both cases, if b 6= 0 the limiting value is finite, independently onwhether W diverges or tends to a constant for N →∞. Consequently, forb 6= 0 these entropies, albeit monotonically increasing functions of W , cannot be extensive.
I A natural question emerges, concerning the kind of systems such type ofentropies could be useful for.
I A possible answer is that the present formalism could be relevantwhenever treating systems highly correlated, where the addition of anew degree of freedom essentially does not change the value of theentropy, for a large number of degrees of freedom.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I One can also consider different scenarios, borrowed from socialsciences, where no thermodynamical or energetical aspects areinvolved, and extensivity is a priori not required. Again suchentropies, that increase very little with the addition of new degreesof freedom, could be play a relevant role in describing situationswhere the amount of information tends to stabilize,irrespectively of the increase of new agents involved in theinformation exchange.
I Consequently, the multiparametric entropy Sa,b,r is compatible withboth scenarios: the standard one, where an increase of the numbersof degrees of freedom converts into an increase of the entropy, andthe “anomalous” one, where an increase of the number of particlesessentially freezes the system, by confining it in the phase space.Excepting for Sq for q > 1, this flexibility in the limit W →∞ isseemingly not shared by the entropies typically used in the literature.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
A new class of trace-form multidimensional entropiesThe reconstruction procedureThe general expressionMain properties: the extensivity problem
I One can also consider different scenarios, borrowed from socialsciences, where no thermodynamical or energetical aspects areinvolved, and extensivity is a priori not required. Again suchentropies, that increase very little with the addition of new degreesof freedom, could be play a relevant role in describing situationswhere the amount of information tends to stabilize,irrespectively of the increase of new agents involved in theinformation exchange.
I Consequently, the multiparametric entropy Sa,b,r is compatible withboth scenarios: the standard one, where an increase of the numbersof degrees of freedom converts into an increase of the entropy, andthe “anomalous” one, where an increase of the number of particlesessentially freezes the system, by confining it in the phase space.Excepting for Sq for q > 1, this flexibility in the limit W →∞ isseemingly not shared by the entropies typically used in the literature.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
A magical recipe to generate multiparametric entropies
P. T., Proc. Royal Society A, to appear (2015).
I Definition 2.9. Let f : [0, 1)→ R be a C∞(0, 1) function,continuous in [0, 1), such that
1− af ′(axσ)− a2xσ σ1+σ f
′′(axσ)
1− af ′(a)> 0 (55)
for x ∈ [0, 1], a ∈ (0, la), σ ∈ (0, lσ), with la, lσ ∈ (0, 1). Then f willbe said to be an admissible function.We shall denote by A the set of admissible functions. The values ofla, lσ usually will depend on the choice of f .
I Definition 5 makes sense for functions f at least of class C2(0, 1).The regularity C∞(0, 1) is required for later convenience.
I Notice that the condition 1− af ′(a) 6= 0 is implicit in Definition 5.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
A magical recipe to generate multiparametric entropies
P. T., Proc. Royal Society A, to appear (2015).
I Definition 2.9. Let f : [0, 1)→ R be a C∞(0, 1) function,continuous in [0, 1), such that
1− af ′(axσ)− a2xσ σ1+σ f
′′(axσ)
1− af ′(a)> 0 (55)
for x ∈ [0, 1], a ∈ (0, la), σ ∈ (0, lσ), with la, lσ ∈ (0, 1). Then f willbe said to be an admissible function.We shall denote by A the set of admissible functions. The values ofla, lσ usually will depend on the choice of f .
I Definition 5 makes sense for functions f at least of class C2(0, 1).The regularity C∞(0, 1) is required for later convenience.
I Notice that the condition 1− af ′(a) 6= 0 is implicit in Definition 5.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
A magical recipe to generate multiparametric entropies
P. T., Proc. Royal Society A, to appear (2015).
I Definition 2.9. Let f : [0, 1)→ R be a C∞(0, 1) function,continuous in [0, 1), such that
1− af ′(axσ)− a2xσ σ1+σ f
′′(axσ)
1− af ′(a)> 0 (55)
for x ∈ [0, 1], a ∈ (0, la), σ ∈ (0, lσ), with la, lσ ∈ (0, 1). Then f willbe said to be an admissible function.We shall denote by A the set of admissible functions. The values ofla, lσ usually will depend on the choice of f .
I Definition 5 makes sense for functions f at least of class C2(0, 1).The regularity C∞(0, 1) is required for later convenience.
I Notice that the condition 1− af ′(a) 6= 0 is implicit in Definition 5.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
I Definition 2.10 Let pii=1,··· ,W , with W ≥ 1,∑W
i=1 pi = 1, be a discreteprobability distribution. Let f be an admissible function. The functional
Sf [p] =kB
1− af ′(a)
W∑i=1
pi
(f (apσi )− pσi
σ+
1− f (a)
σ
)(56)
for a ∈ (0, la), σ ∈ (0, lσ), with la, lσ ∈ (0, 1), will be called the entropyassociated with f .
I Theorem 2.11 For any choice of f ∈ A, Sf [p] satisfies the firstthree SK axioms.
I Theorem 2.12 For any f ∈ A, Sf is weakly composable. In
particular, the function G (t) such that S =∑
i piG(
ln 1pi
)is
G(t) = t − σ a2f ′′(a) + af ′(a)− 1
2(−1 + af ′(a))t2 +
σ2 a3f ′′′(a) + 3a2f ′′(a) + af ′(a)− 1
6(−1 + af ′(a))t3 + ... =
∞∑k=1
γktk
k + 1,
with γ0 = 1, γ1 = σ a2f ′′(a)+af ′(a)−12(1−af ′(a)) , etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
I Definition 2.10 Let pii=1,··· ,W , with W ≥ 1,∑W
i=1 pi = 1, be a discreteprobability distribution. Let f be an admissible function. The functional
Sf [p] =kB
1− af ′(a)
W∑i=1
pi
(f (apσi )− pσi
σ+
1− f (a)
σ
)(56)
for a ∈ (0, la), σ ∈ (0, lσ), with la, lσ ∈ (0, 1), will be called the entropyassociated with f .
I Theorem 2.11 For any choice of f ∈ A, Sf [p] satisfies the firstthree SK axioms.
I Theorem 2.12 For any f ∈ A, Sf is weakly composable. In
particular, the function G (t) such that S =∑
i piG(
ln 1pi
)is
G(t) = t − σ a2f ′′(a) + af ′(a)− 1
2(−1 + af ′(a))t2 +
σ2 a3f ′′′(a) + 3a2f ′′(a) + af ′(a)− 1
6(−1 + af ′(a))t3 + ... =
∞∑k=1
γktk
k + 1,
with γ0 = 1, γ1 = σ a2f ′′(a)+af ′(a)−12(1−af ′(a)) , etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
I Definition 2.10 Let pii=1,··· ,W , with W ≥ 1,∑W
i=1 pi = 1, be a discreteprobability distribution. Let f be an admissible function. The functional
Sf [p] =kB
1− af ′(a)
W∑i=1
pi
(f (apσi )− pσi
σ+
1− f (a)
σ
)(56)
for a ∈ (0, la), σ ∈ (0, lσ), with la, lσ ∈ (0, 1), will be called the entropyassociated with f .
I Theorem 2.11 For any choice of f ∈ A, Sf [p] satisfies the firstthree SK axioms.
I Theorem 2.12 For any f ∈ A, Sf is weakly composable. In
particular, the function G (t) such that S =∑
i piG(
ln 1pi
)is
G(t) = t − σ a2f ′′(a) + af ′(a)− 1
2(−1 + af ′(a))t2 +
σ2 a3f ′′′(a) + 3a2f ′′(a) + af ′(a)− 1
6(−1 + af ′(a))t3 + ... =
∞∑k=1
γktk
k + 1,
with γ0 = 1, γ1 = σ a2f ′′(a)+af ′(a)−12(1−af ′(a)) , etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
We shall briefly describe the thermodynamic properties of the entropiesdiscussed above. Despite of their different functional form, they shareseveral relevant features. Let us first introduce the generalizedlogarithm
Logf (x) :=1
1− af ′(a)
(f (ax−σ)− x−σ
σ+
1− f (a)
σ
). (57)
Equation (56) takes the usual form
Sf [p] = kB
W∑i=1
piLogf
(1
pi
).
In the micro-canonical ensemble, all microstates have equal probabilitypi = 1/W . In this case, Eq. (56) becomes
Sf [W ] := kBLog[f ][W ].
It reduces to the celebrated Boltzmann formula SBG = k lnW whenσ → 0. All the discussion valid for the UGE concerning the existence of aLegendre structure applies also to the class Sf .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
New entropic forms from the recipe
Definition 2.13. The generalized Kaniadakis entropy is the function
S1/x [p] =kB2
W∑i=1
pi
(αp−σi − pσi
σ
). (58)
Indeed, it can be considered as a two-parametric version of Kaniadakisentropy, due to the presence of the extra parameter α = 1/a. It coincideswith the standard Kaniadakis entropy for α = 1 and σ = κ, κ ∈ (−1, 1).The exponential entropy
Sexp[p] :=kB
1− aea
W∑i=1
pi
(eap
σi − pσiσ
+1− ea
σ
)(59)
is defined for a ∈ (0, 1/2), σ ∈ (0, 1/2).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
The sin-entropy is the functional
Ssin[p] :=kB
1− a cos a
W∑i=1
pi
(sin apσi − pσi
σ+
1− sin a
σ
)(60)
with a ∈ (0, 1), σ ∈ (0, 1).
Is it possible to “compose” old entropies to get new ones?For instance, in the theory of modular forms, there is a procedure to passfrom
“old” modular forms → newforms
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
Composition theorem
I Theorem 2.14 Assume that f ∈ A. Let
Sf [p] :=kB
1− a1f ′(a1)
W∑i=1
pi
(f (a1pσ1i )− pσ1i
σ1+
1− f (a1)
σ1
),
with a1 ∈ (0, la1), σ1 ∈ (0, lσ1) be the entropy associated with f .
I Then the function
s(x) :=x
1− a1f ′(a1)
(f (a1xσ1)− xσ1
σ1+
1− f (a1)
σ1
)is an admissible function (it belongs to ∈ A) and
I
Ss [p] :=kB
1− a2s ′(a2)
W∑i=1
pi
(s(a2pσ2i )− pσ2i
σ2+
1− s(a2)
σ2
)(61)
with a2 ∈ (0, la2), σ2 ∈ (0, lσ2), la2 , lσ2 ∈ (0, 1) is also an entropy,associated with s(x), and depending on the four parameters(a1, a2, σ1, σ2).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
Composition theorem
I Theorem 2.14 Assume that f ∈ A. Let
Sf [p] :=kB
1− a1f ′(a1)
W∑i=1
pi
(f (a1pσ1i )− pσ1i
σ1+
1− f (a1)
σ1
),
with a1 ∈ (0, la1), σ1 ∈ (0, lσ1) be the entropy associated with f .
I Then the function
s(x) :=x
1− a1f ′(a1)
(f (a1xσ1)− xσ1
σ1+
1− f (a1)
σ1
)is an admissible function (it belongs to ∈ A) and
I
Ss [p] :=kB
1− a2s ′(a2)
W∑i=1
pi
(s(a2pσ2i )− pσ2i
σ2+
1− s(a2)
σ2
)(61)
with a2 ∈ (0, la2), σ2 ∈ (0, lσ2), la2 , lσ2 ∈ (0, 1) is also an entropy,associated with s(x), and depending on the four parameters(a1, a2, σ1, σ2).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
The microcanonical description and associated thermodynamicsMultiparametric entropies
Composition theorem
I Theorem 2.14 Assume that f ∈ A. Let
Sf [p] :=kB
1− a1f ′(a1)
W∑i=1
pi
(f (a1pσ1i )− pσ1i
σ1+
1− f (a1)
σ1
),
with a1 ∈ (0, la1), σ1 ∈ (0, lσ1) be the entropy associated with f .
I Then the function
s(x) :=x
1− a1f ′(a1)
(f (a1xσ1)− xσ1
σ1+
1− f (a1)
σ1
)is an admissible function (it belongs to ∈ A) and
I
Ss [p] :=kB
1− a2s ′(a2)
W∑i=1
pi
(s(a2pσ2i )− pσ2i
σ2+
1− s(a2)
σ2
)(61)
with a2 ∈ (0, la2), σ2 ∈ (0, lσ2), la2 , lσ2 ∈ (0, 1) is also an entropy,associated with s(x), and depending on the four parameters(a1, a2, σ1, σ2).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
On the asymptotic behaviour of generalized entropies
I From a mathematical point of view, the study of the asymptoticbehaviour of a given entropy, in the limit of large size systems, is not awell defined task, even for the standard Boltzmann-Gibbs entropy.
I Let W be the number of states admitted by a system. We shall focus onthe case of large size systems (which implies W →∞). An entropy isthen a functional S = S [p] defined on the space P∞.
I A simple argument proves that, depending on the choice of thedistribution in P∞, we can get infinitely many different limits even for theSBG case.
Indeed, consider the probability distribution p = (1, 0, 0, . . .) withinfinitely many entries. Then SBG [p] = 0. If p = (1/2, 1/2, 0, 0, . . .),then SBG [p] = ln 2 (in units of kB). Let N = 1080 (i.e., the estimatednumber of atoms in the observable Universe), andp = (1/N , 1/N , . . . , 1/N︸ ︷︷ ︸
N−times
, 0, 0, . . .), then SBG [p] = 80 · ln 10 ' 184.2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
On the asymptotic behaviour of generalized entropies
I From a mathematical point of view, the study of the asymptoticbehaviour of a given entropy, in the limit of large size systems, is not awell defined task, even for the standard Boltzmann-Gibbs entropy.
I Let W be the number of states admitted by a system. We shall focus onthe case of large size systems (which implies W →∞). An entropy isthen a functional S = S [p] defined on the space P∞.
I A simple argument proves that, depending on the choice of thedistribution in P∞, we can get infinitely many different limits even for theSBG case.
Indeed, consider the probability distribution p = (1, 0, 0, . . .) withinfinitely many entries. Then SBG [p] = 0. If p = (1/2, 1/2, 0, 0, . . .),then SBG [p] = ln 2 (in units of kB). Let N = 1080 (i.e., the estimatednumber of atoms in the observable Universe), andp = (1/N , 1/N , . . . , 1/N︸ ︷︷ ︸
N−times
, 0, 0, . . .), then SBG [p] = 80 · ln 10 ' 184.2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
On the asymptotic behaviour of generalized entropies
I From a mathematical point of view, the study of the asymptoticbehaviour of a given entropy, in the limit of large size systems, is not awell defined task, even for the standard Boltzmann-Gibbs entropy.
I Let W be the number of states admitted by a system. We shall focus onthe case of large size systems (which implies W →∞). An entropy isthen a functional S = S [p] defined on the space P∞.
I A simple argument proves that, depending on the choice of thedistribution in P∞, we can get infinitely many different limits even for theSBG case.
Indeed, consider the probability distribution p = (1, 0, 0, . . .) withinfinitely many entries. Then SBG [p] = 0. If p = (1/2, 1/2, 0, 0, . . .),then SBG [p] = ln 2 (in units of kB). Let N = 1080 (i.e., the estimatednumber of atoms in the observable Universe), andp = (1/N , 1/N , . . . , 1/N︸ ︷︷ ︸
N−times
, 0, 0, . . .), then SBG [p] = 80 · ln 10 ' 184.2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I From the point of view of probability and information theory, there is noway to have uniqueness of the limit on the full space P∞.
I In the domain of classical thermodynamics, a priori the same objectionapplies. However, if we accept to restrict to the important case of theuniform distribution (which is not the only physically interesting case),then one can define properly a thermodynamic limit of an entropy onthe uniform distribution. From a physical perspective, this wouldcorrespond to the case of a double limit, both of large system size and oflarge times.
I The only meaningful way to compare the behaviour of different entropiesis to compute them all in the same regime. Once we accept to restrict touniform probabilities, then the known entropies assume the form of thefollowing asymptotic functions: either (lnW )a or W b (possibly multipliedby parameters), or the product of these forms. Presently, we cannotexclude that other forms could also be possible.
This is a consequence of the analysis of scaling laws performed in R.Hanel and S. Thurner, Europhys. Lett. 93 20006 (2011). .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I From the point of view of probability and information theory, there is noway to have uniqueness of the limit on the full space P∞.
I In the domain of classical thermodynamics, a priori the same objectionapplies. However, if we accept to restrict to the important case of theuniform distribution (which is not the only physically interesting case),then one can define properly a thermodynamic limit of an entropy onthe uniform distribution. From a physical perspective, this wouldcorrespond to the case of a double limit, both of large system size and oflarge times.
I The only meaningful way to compare the behaviour of different entropiesis to compute them all in the same regime. Once we accept to restrict touniform probabilities, then the known entropies assume the form of thefollowing asymptotic functions: either (lnW )a or W b (possibly multipliedby parameters), or the product of these forms. Presently, we cannotexclude that other forms could also be possible.
This is a consequence of the analysis of scaling laws performed in R.Hanel and S. Thurner, Europhys. Lett. 93 20006 (2011). .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I From the point of view of probability and information theory, there is noway to have uniqueness of the limit on the full space P∞.
I In the domain of classical thermodynamics, a priori the same objectionapplies. However, if we accept to restrict to the important case of theuniform distribution (which is not the only physically interesting case),then one can define properly a thermodynamic limit of an entropy onthe uniform distribution. From a physical perspective, this wouldcorrespond to the case of a double limit, both of large system size and oflarge times.
I The only meaningful way to compare the behaviour of different entropiesis to compute them all in the same regime. Once we accept to restrict touniform probabilities, then the known entropies assume the form of thefollowing asymptotic functions: either (lnW )a or W b (possibly multipliedby parameters), or the product of these forms. Presently, we cannotexclude that other forms could also be possible.
This is a consequence of the analysis of scaling laws performed in R.Hanel and S. Thurner, Europhys. Lett. 93 20006 (2011). .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I The problem of “comparing” entropies in the large size limit finds its mostsimple answer in the regime, if exists, where they are extensive. Inclassical thermodynamics, the only reason to consider generalizedentropies is the fact that they can be extensive in regimes where the SBG
entropy is not. This is the truly important limiting property.
I What is crucial is that all admissible entropies have the same behaviorin the regime where they are extensive, i.e. proportional to the number Nof particles of the system.
I Therefore, it is not surprising that different entropies could share the sameasymptotic behavior.
When we are interested in more general contexts as probability theory andinformation theory, then all of the infinitely many possible distributions area priori relevant. Therefore, according to the previous discussion, thecomparison among asymptotic behaviors of different entropies loses itsmeaning.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I The problem of “comparing” entropies in the large size limit finds its mostsimple answer in the regime, if exists, where they are extensive. Inclassical thermodynamics, the only reason to consider generalizedentropies is the fact that they can be extensive in regimes where the SBG
entropy is not. This is the truly important limiting property.
I What is crucial is that all admissible entropies have the same behaviorin the regime where they are extensive, i.e. proportional to the number Nof particles of the system.
I Therefore, it is not surprising that different entropies could share the sameasymptotic behavior.
When we are interested in more general contexts as probability theory andinformation theory, then all of the infinitely many possible distributions area priori relevant. Therefore, according to the previous discussion, thecomparison among asymptotic behaviors of different entropies loses itsmeaning.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
I The problem of “comparing” entropies in the large size limit finds its mostsimple answer in the regime, if exists, where they are extensive. Inclassical thermodynamics, the only reason to consider generalizedentropies is the fact that they can be extensive in regimes where the SBG
entropy is not. This is the truly important limiting property.
I What is crucial is that all admissible entropies have the same behaviorin the regime where they are extensive, i.e. proportional to the number Nof particles of the system.
I Therefore, it is not surprising that different entropies could share the sameasymptotic behavior.
When we are interested in more general contexts as probability theory andinformation theory, then all of the infinitely many possible distributions area priori relevant. Therefore, according to the previous discussion, thecomparison among asymptotic behaviors of different entropies loses itsmeaning.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
State of Art
I The correspondence among groups and entropies leads naturally to thenotion of universal-group entropy SU , as the entropy associated with theLazard universal formal group.
I It includes a huge class of entropies, satisfying the first three SK axioms,and the composability axiom at least in the weak sense.
I The Legendre structure of the group entropies is not recovered fromclassical constraints; non standard ones should be introduced (openproblem).
I A simple reconstruction procedure can be defined, allowing to define thetrace-form entropy associated with a group law.
I Multiparametric entropies can be introduced from a “recipe”; newentropies can be generated from old ones.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
State of Art
I The correspondence among groups and entropies leads naturally to thenotion of universal-group entropy SU , as the entropy associated with theLazard universal formal group.
I It includes a huge class of entropies, satisfying the first three SK axioms,and the composability axiom at least in the weak sense.
I The Legendre structure of the group entropies is not recovered fromclassical constraints; non standard ones should be introduced (openproblem).
I A simple reconstruction procedure can be defined, allowing to define thetrace-form entropy associated with a group law.
I Multiparametric entropies can be introduced from a “recipe”; newentropies can be generated from old ones.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
State of Art
I The correspondence among groups and entropies leads naturally to thenotion of universal-group entropy SU , as the entropy associated with theLazard universal formal group.
I It includes a huge class of entropies, satisfying the first three SK axioms,and the composability axiom at least in the weak sense.
I The Legendre structure of the group entropies is not recovered fromclassical constraints; non standard ones should be introduced (openproblem).
I A simple reconstruction procedure can be defined, allowing to define thetrace-form entropy associated with a group law.
I Multiparametric entropies can be introduced from a “recipe”; newentropies can be generated from old ones.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
State of Art
I The correspondence among groups and entropies leads naturally to thenotion of universal-group entropy SU , as the entropy associated with theLazard universal formal group.
I It includes a huge class of entropies, satisfying the first three SK axioms,and the composability axiom at least in the weak sense.
I The Legendre structure of the group entropies is not recovered fromclassical constraints; non standard ones should be introduced (openproblem).
I A simple reconstruction procedure can be defined, allowing to define thetrace-form entropy associated with a group law.
I Multiparametric entropies can be introduced from a “recipe”; newentropies can be generated from old ones.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
State of Art
I The correspondence among groups and entropies leads naturally to thenotion of universal-group entropy SU , as the entropy associated with theLazard universal formal group.
I It includes a huge class of entropies, satisfying the first three SK axioms,and the composability axiom at least in the weak sense.
I The Legendre structure of the group entropies is not recovered fromclassical constraints; non standard ones should be introduced (openproblem).
I A simple reconstruction procedure can be defined, allowing to define thetrace-form entropy associated with a group law.
I Multiparametric entropies can be introduced from a “recipe”; newentropies can be generated from old ones.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
Generalized Entropy
Lazard’s universal
formal group
and
universal-group
Entropy
Generalized Entropy Group law
(weak or strong sense)
Direct Approach Inverse Approach
Group law
Composability
axiom
Reconstruction
procedure
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
The Universal Group EntropyDistribution functions and thermodynamic properties
An entropy for the rational group lawA “magical recipe” to generate multiparametric entropies
On the asymptotic behaviour of generalized entropiesState of Art
Thank you!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
GROUPS, ENTROPIES AND NUMBERTHEORY
Piergiulio Tempesta
Universidad Complutense de Madridand
Instituto de Ciencias Matematicas (ICMAT),Madrid, Spain.
TEMPLETON SCHOOL ON FOUNDATIONS OF COMPLEXITY
October 14 - 15, 2015
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
LECTURE III
Information Theory of group entropies:Renyi’s entropy and
the Z-class of Entropies
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
OutlineInformation-theoretical aspects of the notion of entropy
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
The Z-class of EntropiesThe composability problem revisitedThe Z-entropies: main definitions
Main properties of the Z-entropiesConcavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-MittalentropiesZ-delta entropiesInformation-theoretical content: Z-divergences
State of ArtPiergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.
I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.
I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Information Theory and Entropy
I H. Nyquist, Bell Syst. Tech. J., 3, 324 (1924).R. V. Hartley, Bell Syst. Tech. J. 7, 535 (1928).
I A. Kolmogorov, Atti della R. Acc. Naz. Lincei, 12, 388 (1930)M. Nagumo, Jap. J. Math. 7, 71 (1930)
I C. Shannon, Bell Syst. Tech. J. 27, 379 (1948); 27, 623 (1948),Shannon, W. Weaver, The mathematical Theory of Communication,University of Illinois Press, Urbana, IL, 1949.
I A. I. Khinchin, Mathematical Foundations of Information Theory,Dover, New York, 1957.
I E. T. Jaynes, Information theory and Statistical Mechanics. Part I,Phys. Rev. 106, 620 (1957); Part II, Phys. Rev. 108, 171 (1957).
I S. Kullback, Information Theory and Statistics, Wiley, 1959.I A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.I B. B. Mandelbrot, Fractal-form. Chance and Dimension, Freeman,
1977.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Modern Theory: Applicability
I From the original vision of Shannon, concerning the limits on signalprocessing operations, i.e. storing, compressing andcommunicating data (e.g. the “noisy channel coding theorem”),Information Theory has broadened to cover many different areas:
I Cryptography
I Statistical Inference
I Quantum Computing
I Pattern Recognition
I Theoretical Linguistics
I Ecology, etc.
I A crucial notion is that of Information Entropy.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Some foundational aspects
A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.
Let I be the amount of information (expressed in bits) associated withan event, described by a random variable X , whose probabilitydistribution is p := piWi=1.We shall make three fundamental assumptions.
I 1) Information should depends purely on p.
I 2) Information should be additive for two independent events.It means that if we observe the outcome of two independent eventswith probabilities p and q, the total information associated is thesum of the two. This implies the Cauchy functional equation
I(p · q) = I(p) + I(q).
I Under very mild assumptions, this equation admits a unique class ofsolutions: I(p) = −k ln2(p). For instance, by putting I (1/2) = 1,we get the so called Hartley measure of information: I = − ln2(p)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Some foundational aspects
A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.
Let I be the amount of information (expressed in bits) associated withan event, described by a random variable X , whose probabilitydistribution is p := piWi=1.We shall make three fundamental assumptions.
I 1) Information should depends purely on p.I 2) Information should be additive for two independent events.
It means that if we observe the outcome of two independent eventswith probabilities p and q, the total information associated is thesum of the two. This implies the Cauchy functional equation
I(p · q) = I(p) + I(q).
I Under very mild assumptions, this equation admits a unique class ofsolutions: I(p) = −k ln2(p). For instance, by putting I (1/2) = 1,we get the so called Hartley measure of information: I = − ln2(p)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Some foundational aspects
A. Renyi, Probability Theory, North–Holland, Amsterdam, 1970.
Let I be the amount of information (expressed in bits) associated withan event, described by a random variable X , whose probabilitydistribution is p := piWi=1.We shall make three fundamental assumptions.
I 1) Information should depends purely on p.I 2) Information should be additive for two independent events.
It means that if we observe the outcome of two independent eventswith probabilities p and q, the total information associated is thesum of the two. This implies the Cauchy functional equation
I(p · q) = I(p) + I(q).
I Under very mild assumptions, this equation admits a unique class ofsolutions: I(p) = −k ln2(p). For instance, by putting I (1/2) = 1,we get the so called Hartley measure of information: I = − ln2(p)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
P. Jizba, T. Arimitsu, Ann. of Phys. 312, 17 (2004).
I 3) If different amounts of information occur with differentprobabilities, the total amount of information is the average ofthe individual information, weighted by the probabilities of theiroccurrences.
Let A1, . . . ,An be n independent random variables, interpreted aspossible outcomes of an experiment, with probabilities p1, . . . , pn.Assume that Ak conveys Ik bits of information. Then, a natural wayto express the total amount of informationI = I(p1, . . . , pn; I1, . . . , In) is
I =n∑
k=1
pkIk .
I Notice that this linear averaging is just a particular case of a moregeneral mean, introduced by Kolmogorov and Nagumo in 1930!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
P. Jizba, T. Arimitsu, Ann. of Phys. 312, 17 (2004).
I 3) If different amounts of information occur with differentprobabilities, the total amount of information is the average ofthe individual information, weighted by the probabilities of theiroccurrences.
Let A1, . . . ,An be n independent random variables, interpreted aspossible outcomes of an experiment, with probabilities p1, . . . , pn.Assume that Ak conveys Ik bits of information. Then, a natural wayto express the total amount of informationI = I(p1, . . . , pn; I1, . . . , In) is
I =n∑
k=1
pkIk .
I Notice that this linear averaging is just a particular case of a moregeneral mean, introduced by Kolmogorov and Nagumo in 1930!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
The Kolmogorov-Nagumo approach
Kolmogorov and Nagumo independently introduced the notion ofKolmogorov-Nagumo (KN) average.Definition 3.1. Let f be an arbitrary real function, and piWi=1 be a probabilitydistribution. Let fk := f (k), where k ∈ N. Assume that φ is a monotonicallyincreasing real function, called the KN function. The KN average of f is
〈f 〉 = φ−1
(W∑k=0
pkφ(fk)
)Now, fix an additional monotonically increasing function ω(x). We shall callinformation content Ik the quantity
Ik = ω
(1
pk
).
Therefore, the average information content is
〈I 〉 = φ−1
(W∑k=0
pkφ
(ω
(1
pk
)))Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Without loss of generality, we shall define the information content by therelation ω(x) = ln2(x). We are led to the most general measure of theamount of transmitted information:
〈I 〉 = φ−1
(W∑k=0
pkφ
(ln2
1
pk
))
Problem. What are the mathematical forms allowed for the KN functionφ(x)? Once again, the composition process plays a crucial role.Consider an experiment E which comes from the union of two
independent experiments E1 and E2. Suppose that I(1)k bits of
information are received with probability pk , k = 1, . . . , n from the
experiment E1 and I(2)l bits of information are received with probability
ql from the experiment E2, l = 1, . . . ,m.From the postulate of additivity, we get
φ−1
(n∑
k=1
m∑l=1
pkqlφ(I(1)k + I(2)
l
))= φ−1
(n∑
k=1
pkφ(I(1)k
))+φ−1
(m∑l=1
qlφ(I(2)l
))
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Shannon’s and Renyi entropies
I Result. The additivity postulate allows only for two classes: f linearand f exponential.a) φ(x) = cx (the most elementary KN function).
I We obtain
〈I〉 = −n∑
k=1
pk log2(pk)
i.e., the celebrated Shannon entropyI b) φ(x) = 2(1−α)x−1
γ .I We obtain
〈I〉 = Hα(p) :=1
1− αLog2
(n∑
k=1
pαk
).
It is called the Renyi entropy or information measure of order α.I Renyi entropy has been axiomatized by several authors: Renyi
himself, Daroczy (1964), Jizba and Arimitsu (2004).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Shannon’s and Renyi entropies
I Result. The additivity postulate allows only for two classes: f linearand f exponential.a) φ(x) = cx (the most elementary KN function).
I We obtain
〈I〉 = −n∑
k=1
pk log2(pk)
i.e., the celebrated Shannon entropy
I b) φ(x) = 2(1−α)x−1γ .
I We obtain
〈I〉 = Hα(p) :=1
1− αLog2
(n∑
k=1
pαk
).
It is called the Renyi entropy or information measure of order α.I Renyi entropy has been axiomatized by several authors: Renyi
himself, Daroczy (1964), Jizba and Arimitsu (2004).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Shannon’s and Renyi entropies
I Result. The additivity postulate allows only for two classes: f linearand f exponential.a) φ(x) = cx (the most elementary KN function).
I We obtain
〈I〉 = −n∑
k=1
pk log2(pk)
i.e., the celebrated Shannon entropyI b) φ(x) = 2(1−α)x−1
γ .
I We obtain
〈I〉 = Hα(p) :=1
1− αLog2
(n∑
k=1
pαk
).
It is called the Renyi entropy or information measure of order α.I Renyi entropy has been axiomatized by several authors: Renyi
himself, Daroczy (1964), Jizba and Arimitsu (2004).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Shannon’s and Renyi entropies
I Result. The additivity postulate allows only for two classes: f linearand f exponential.a) φ(x) = cx (the most elementary KN function).
I We obtain
〈I〉 = −n∑
k=1
pk log2(pk)
i.e., the celebrated Shannon entropyI b) φ(x) = 2(1−α)x−1
γ .I We obtain
〈I〉 = Hα(p) :=1
1− αLog2
(n∑
k=1
pαk
).
It is called the Renyi entropy or information measure of order α.
I Renyi entropy has been axiomatized by several authors: Renyihimself, Daroczy (1964), Jizba and Arimitsu (2004).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Shannon’s and Renyi entropies
I Result. The additivity postulate allows only for two classes: f linearand f exponential.a) φ(x) = cx (the most elementary KN function).
I We obtain
〈I〉 = −n∑
k=1
pk log2(pk)
i.e., the celebrated Shannon entropyI b) φ(x) = 2(1−α)x−1
γ .I We obtain
〈I〉 = Hα(p) :=1
1− αLog2
(n∑
k=1
pαk
).
It is called the Renyi entropy or information measure of order α.I Renyi entropy has been axiomatized by several authors: Renyi
himself, Daroczy (1964), Jizba and Arimitsu (2004).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Consequently, Information Entropy is the expected value(average) of the information contained in an experiment (message,flow of information, etc.).
I Given a random variable X , with associated a certain distributionof outcomes x1, . . . , xn, information entropy can be interpreted asa measure of the amount of uncertainty associated with the value ofX if only the distribution is known.
I If the distribution associated with X is constant, i.e. equal to someknown value with probability 1, then the information entropy isminimal, and equal to zero.
I If X takes a (finite) number of values, each of them equally likely,(i.e. they are uniformly distributed), the information entropy ismaximal.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Consequently, Information Entropy is the expected value(average) of the information contained in an experiment (message,flow of information, etc.).
I Given a random variable X , with associated a certain distributionof outcomes x1, . . . , xn, information entropy can be interpreted asa measure of the amount of uncertainty associated with the value ofX if only the distribution is known.
I If the distribution associated with X is constant, i.e. equal to someknown value with probability 1, then the information entropy isminimal, and equal to zero.
I If X takes a (finite) number of values, each of them equally likely,(i.e. they are uniformly distributed), the information entropy ismaximal.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Consequently, Information Entropy is the expected value(average) of the information contained in an experiment (message,flow of information, etc.).
I Given a random variable X , with associated a certain distributionof outcomes x1, . . . , xn, information entropy can be interpreted asa measure of the amount of uncertainty associated with the value ofX if only the distribution is known.
I If the distribution associated with X is constant, i.e. equal to someknown value with probability 1, then the information entropy isminimal, and equal to zero.
I If X takes a (finite) number of values, each of them equally likely,(i.e. they are uniformly distributed), the information entropy ismaximal.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Consequently, Information Entropy is the expected value(average) of the information contained in an experiment (message,flow of information, etc.).
I Given a random variable X , with associated a certain distributionof outcomes x1, . . . , xn, information entropy can be interpreted asa measure of the amount of uncertainty associated with the value ofX if only the distribution is known.
I If the distribution associated with X is constant, i.e. equal to someknown value with probability 1, then the information entropy isminimal, and equal to zero.
I If X takes a (finite) number of values, each of them equally likely,(i.e. they are uniformly distributed), the information entropy ismaximal.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Examples. Suppose to transmit a string M of bits (0s and 1s).
I If the receiver already knows the value of each bit of M, (e.g.p(0) = 1, p(1) = 0, no information is transmitted. In this case,S = 0
I If each bit is independently equally likely transmitted (i.e.p(0) = 1/2, p(1) = 1/2), then we have transmitted M shannons ofinformation. In this case, S = log2M.
I In the intermediate cases, the Shannon information entropy takesvalues between 0 and log2M.
I More generally, if a random variable X takes values x1, . . . , xn,where xi ∈ X, the Shannon information entropy takes values
S(M) = −∑xi∈X
p(xi )log2(p(xi )).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Examples. Suppose to transmit a string M of bits (0s and 1s).
I If the receiver already knows the value of each bit of M, (e.g.p(0) = 1, p(1) = 0, no information is transmitted. In this case,S = 0
I If each bit is independently equally likely transmitted (i.e.p(0) = 1/2, p(1) = 1/2), then we have transmitted M shannons ofinformation. In this case, S = log2M.
I In the intermediate cases, the Shannon information entropy takesvalues between 0 and log2M.
I More generally, if a random variable X takes values x1, . . . , xn,where xi ∈ X, the Shannon information entropy takes values
S(M) = −∑xi∈X
p(xi )log2(p(xi )).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Examples. Suppose to transmit a string M of bits (0s and 1s).
I If the receiver already knows the value of each bit of M, (e.g.p(0) = 1, p(1) = 0, no information is transmitted. In this case,S = 0
I If each bit is independently equally likely transmitted (i.e.p(0) = 1/2, p(1) = 1/2), then we have transmitted M shannons ofinformation. In this case, S = log2M.
I In the intermediate cases, the Shannon information entropy takesvalues between 0 and log2M.
I More generally, if a random variable X takes values x1, . . . , xn,where xi ∈ X, the Shannon information entropy takes values
S(M) = −∑xi∈X
p(xi )log2(p(xi )).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Examples. Suppose to transmit a string M of bits (0s and 1s).
I If the receiver already knows the value of each bit of M, (e.g.p(0) = 1, p(1) = 0, no information is transmitted. In this case,S = 0
I If each bit is independently equally likely transmitted (i.e.p(0) = 1/2, p(1) = 1/2), then we have transmitted M shannons ofinformation. In this case, S = log2M.
I In the intermediate cases, the Shannon information entropy takesvalues between 0 and log2M.
I More generally, if a random variable X takes values x1, . . . , xn,where xi ∈ X, the Shannon information entropy takes values
S(M) = −∑xi∈X
p(xi )log2(p(xi )).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
On the notion of Information Entropy
I Examples. Suppose to transmit a string M of bits (0s and 1s).
I If the receiver already knows the value of each bit of M, (e.g.p(0) = 1, p(1) = 0, no information is transmitted. In this case,S = 0
I If each bit is independently equally likely transmitted (i.e.p(0) = 1/2, p(1) = 1/2), then we have transmitted M shannons ofinformation. In this case, S = log2M.
I In the intermediate cases, the Shannon information entropy takesvalues between 0 and log2M.
I More generally, if a random variable X takes values x1, . . . , xn,where xi ∈ X, the Shannon information entropy takes values
S(M) = −∑xi∈X
p(xi )log2(p(xi )).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
The Renyi entropy, revisited
A. Renyi, On measures of information and entropy, Proc. of the 4th BerkeleySymposium on Mathematics, Statistics and Probability, 547, 1960. Thecelebrated Renyi entropy is one of most important measure entropies.
Hα(p1, . . . , pW ) :=ln(∑W
i=1 pαi
)1− α .
It is additive:Hα(A ∪ B) = Hα(A) + Hα(B).
It is concave for 0 < α < 1. In the limit α→ 1, it reduces to Shannon’sentropy. For α = 0, it reduces to Hartley’s entropy:
H0 = ln W .
The differential Renyi entropy of a distribution P with density p is given by
hα(P) =1
1− α ln
∫(p(x))αdx .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Renyi divergence
T. van Erven, P. Harremoes, arXiv: 1206.2459v2 (2014).To this aim, let (X ,Σ, µ) be a measure space, where X is a set, Σ a σ–algebraover X , and µ : Σ→ R+ ∪ 0 a measure. A Σ–measurable functionp : X → R+ ∪ 0 will be called a probability distribution function (p.d.f.) if∫
X
pdµ = 1. (1)
The probability measure induced by a p.d.f. p is defined by
P(E) =
∫E
p(x)dµ(x), ∀E ∈ Σ. (2)
We shall assume that (X ,Σ, µ) is a σ–finite measure space.A measure P is called absolutely continuous with respect to a measure Q ifP(A) = 0 whenever Q(A) = 0 for all events A ∈ Σ. We shall write P Q if Pis absolutely continuous with respect to Q.We shall assume that our probability measures are absolutely continuous withrespect to µ. We will also identify functions differing on a µ–null set only.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Definition 3.1. The Renyi divergence of order α of P from Q is defined to be
Dα(P‖Q) :=1
α− 1ln
(∫pαi
qα−1
)dµ,
where we adopt the conventions that 0/0 = 0 and x/0 =∞ for x > 0.Properties.
D1(P‖Q) = DKL(P‖Q),
i.e. it coincides with the celebrated Kullback-Leibler divergence.Kullback-Leibler divergence:
D1(P‖Q) :=
∫p ln
p
qdµ.
Property 3.2 For any order α ∈ [0,∞],
Dα(P‖Q) ≥ 0.
For α > 0, we have
Dα(P‖Q) = 0 if and only if P = Q.
For α = 0,Dα(P‖Q) = 0 if and only if Q P.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Property 3.3 (joint convexity). For any order α ∈ [0, 1], and for any twopairs of probability measures (P0,Q0) and (P1,Q1), and any 0 < λ < 1,we have
Dα [(1− λ)P0 + λP1 ‖ (1− λ)Q0 + λQ1] ≤ (1−λ)Dα (P0 ‖ Q0)+λDα (P1 ‖ Q1)
I Property 3.4 (convexity in the second argument).
for any order α ∈ [0,∞] Renyi divergence is convex in its secondargument: for any probability distributions P, Q0 and Q1 we have
Dα [P ‖ (1− λ)Q0 + λQ1] ≤ (1− λ)Dα (P ‖ Q0) + λDα (P ‖ Q1) .
I Property 3.5 For any 0 < α ≤ β < 1,
α
β
1− β1− αDβ(P ‖ Q) ≤ Dα(P ‖ Q) ≤ Dβ(P ‖ Q).
This means that the topologies induced by Renyi divergences of orderα ∈ (0, 1) are all equivalent.
I Property 3.6 (Skew symmetry) For any 0 < α < 1,
Dα(P ‖ Q) =α
1− αD1−α(Q ‖ P).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Property 3.3 (joint convexity). For any order α ∈ [0, 1], and for any twopairs of probability measures (P0,Q0) and (P1,Q1), and any 0 < λ < 1,we have
Dα [(1− λ)P0 + λP1 ‖ (1− λ)Q0 + λQ1] ≤ (1−λ)Dα (P0 ‖ Q0)+λDα (P1 ‖ Q1)
I Property 3.4 (convexity in the second argument).
for any order α ∈ [0,∞] Renyi divergence is convex in its secondargument: for any probability distributions P, Q0 and Q1 we have
Dα [P ‖ (1− λ)Q0 + λQ1] ≤ (1− λ)Dα (P ‖ Q0) + λDα (P ‖ Q1) .
I Property 3.5 For any 0 < α ≤ β < 1,
α
β
1− β1− αDβ(P ‖ Q) ≤ Dα(P ‖ Q) ≤ Dβ(P ‖ Q).
This means that the topologies induced by Renyi divergences of orderα ∈ (0, 1) are all equivalent.
I Property 3.6 (Skew symmetry) For any 0 < α < 1,
Dα(P ‖ Q) =α
1− αD1−α(Q ‖ P).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Property 3.3 (joint convexity). For any order α ∈ [0, 1], and for any twopairs of probability measures (P0,Q0) and (P1,Q1), and any 0 < λ < 1,we have
Dα [(1− λ)P0 + λP1 ‖ (1− λ)Q0 + λQ1] ≤ (1−λ)Dα (P0 ‖ Q0)+λDα (P1 ‖ Q1)
I Property 3.4 (convexity in the second argument).
for any order α ∈ [0,∞] Renyi divergence is convex in its secondargument: for any probability distributions P, Q0 and Q1 we have
Dα [P ‖ (1− λ)Q0 + λQ1] ≤ (1− λ)Dα (P ‖ Q0) + λDα (P ‖ Q1) .
I Property 3.5 For any 0 < α ≤ β < 1,
α
β
1− β1− αDβ(P ‖ Q) ≤ Dα(P ‖ Q) ≤ Dβ(P ‖ Q).
This means that the topologies induced by Renyi divergences of orderα ∈ (0, 1) are all equivalent.
I Property 3.6 (Skew symmetry) For any 0 < α < 1,
Dα(P ‖ Q) =α
1− αD1−α(Q ‖ P).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Property 3.3 (joint convexity). For any order α ∈ [0, 1], and for any twopairs of probability measures (P0,Q0) and (P1,Q1), and any 0 < λ < 1,we have
Dα [(1− λ)P0 + λP1 ‖ (1− λ)Q0 + λQ1] ≤ (1−λ)Dα (P0 ‖ Q0)+λDα (P1 ‖ Q1)
I Property 3.4 (convexity in the second argument).
for any order α ∈ [0,∞] Renyi divergence is convex in its secondargument: for any probability distributions P, Q0 and Q1 we have
Dα [P ‖ (1− λ)Q0 + λQ1] ≤ (1− λ)Dα (P ‖ Q0) + λDα (P ‖ Q1) .
I Property 3.5 For any 0 < α ≤ β < 1,
α
β
1− β1− αDβ(P ‖ Q) ≤ Dα(P ‖ Q) ≤ Dβ(P ‖ Q).
This means that the topologies induced by Renyi divergences of orderα ∈ (0, 1) are all equivalent.
I Property 3.6 (Skew symmetry) For any 0 < α < 1,
Dα(P ‖ Q) =α
1− αD1−α(Q ‖ P).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Lesche stability of Renyi entropy
B. Lesche, J. Stat. Phys. 27, 419 (1982).
I Lesche’s stability. Let f (x) be a scalar quantity, x ∈ X ⊂ Rn be astate variable. Lesche’s criterion provides a necessary condition forf (x) to be observable.
I Let
‖ x− x′ ‖1=n∑k
| xk − x′
k |
be the Holder l1 metric on Rn. Then, ∀ε > 0, there exists δε > 0such that, for any pair x, x′, one has
‖ x− x‘ ‖≤ δε =⇒ | f (x)− f (x′) |fmax
< ε
I This property is nothing but the uniform continuity of f (x) on thestate space X .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Lesche stability of Renyi entropy
B. Lesche, J. Stat. Phys. 27, 419 (1982).
I Lesche’s stability. Let f (x) be a scalar quantity, x ∈ X ⊂ Rn be astate variable. Lesche’s criterion provides a necessary condition forf (x) to be observable.
I Let
‖ x− x′ ‖1=n∑k
| xk − x′
k |
be the Holder l1 metric on Rn. Then, ∀ε > 0, there exists δε > 0such that, for any pair x, x′, one has
‖ x− x‘ ‖≤ δε =⇒ | f (x)− f (x′) |fmax
< ε
I This property is nothing but the uniform continuity of f (x) on thestate space X .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Lesche stability of Renyi entropy
B. Lesche, J. Stat. Phys. 27, 419 (1982).
I Lesche’s stability. Let f (x) be a scalar quantity, x ∈ X ⊂ Rn be astate variable. Lesche’s criterion provides a necessary condition forf (x) to be observable.
I Let
‖ x− x′ ‖1=n∑k
| xk − x′
k |
be the Holder l1 metric on Rn. Then, ∀ε > 0, there exists δε > 0such that, for any pair x, x′, one has
‖ x− x‘ ‖≤ δε =⇒ | f (x)− f (x′) |fmax
< ε
I This property is nothing but the uniform continuity of f (x) on thestate space X .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Many of the trace-form entropies are Lesche stable. In particular,Tsallis’ entropy and Kaniadakis’ entropy are.There are results stating that, under mild conditions, a very largeclass of trace-form entropies are indeed Lesche stable.For long time, Renyi entropy was considered to be not Lesche stable.However, this is not really the case.
I P. Jizba and T. Arimitsu, PRE 69, 026128 (2004).Results.a) Renyi’s entropy is Lesche stable for systems with a finite numberof microstates.b) For systems with an infinite number of microstates, the domainof instability has zero Bhattacharyya measure. The possibleinstabilities can be emended by introducing a coarse graining into anactual measurement.c) In case of systems with continuous probability distributions ormultifractal systems, again the lesche condition applies.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Many of the trace-form entropies are Lesche stable. In particular,Tsallis’ entropy and Kaniadakis’ entropy are.There are results stating that, under mild conditions, a very largeclass of trace-form entropies are indeed Lesche stable.For long time, Renyi entropy was considered to be not Lesche stable.However, this is not really the case.
I P. Jizba and T. Arimitsu, PRE 69, 026128 (2004).Results.a) Renyi’s entropy is Lesche stable for systems with a finite numberof microstates.b) For systems with an infinite number of microstates, the domainof instability has zero Bhattacharyya measure. The possibleinstabilities can be emended by introducing a coarse graining into anactual measurement.c) In case of systems with continuous probability distributions ormultifractal systems, again the lesche condition applies.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Einstein’s likelihood principle
A. Einstein, The Theory of the Opalescence of Homogeneous Fluids andLiquid Mixtures near the Critical State, Annalen der Physik 33,1275-1298 (1910).E. G. D. Cohen, Pramana J. of Physics, 64, 635 (2005).C. Tsallis and H. J. Haubold, EPL, 110, 30005 (2015).
A longstanding unsolved problem. The foundations of statisticalmechanics should be based on a probabilistic approach or on a dynamicalone?Boltzmann’s approach at the equilibrium: SBG = k ln W , where W is thetotal amount of microscopic possibilities, assumed equally probable.
Einstein’s point of view was a dynamical one: “one cannot fix theweights of the various regions in phase space without using the dynamics,i.e. the equations of motion of the system” (Cohen).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Therefore, Einstein never used Boltzmann’s principle. Instead, heproposed the equation (the reversal of Boltzmann’s formula)
W ∝ eSBG /k .
The meaning of this formula is that W is determined from theentropy S , which, as any other physical quantity, is determinedfrom the dynamics. We shall call W the likelihood function.
I Now, assume that we have two systems A and B which arestatistically (and dynamically) independent. Then we postulate that
W(A ∪ B) =W(A)W(B).
Following Tsallis and Haubold, we shall call it the Einsteinlikelihood principle.This principle has very profound consequences. It implies that thephysics of the system A ∪ B depends on the systems A and B only,without any need for the knowledge of the rest of the Universe.Needless to say, A and B are allowed to stay in an arbitrary (a prioriunknown) state.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Therefore, Einstein never used Boltzmann’s principle. Instead, heproposed the equation (the reversal of Boltzmann’s formula)
W ∝ eSBG /k .
The meaning of this formula is that W is determined from theentropy S , which, as any other physical quantity, is determinedfrom the dynamics. We shall call W the likelihood function.
I Now, assume that we have two systems A and B which arestatistically (and dynamically) independent. Then we postulate that
W(A ∪ B) =W(A)W(B).
Following Tsallis and Haubold, we shall call it the Einsteinlikelihood principle.This principle has very profound consequences. It implies that thephysics of the system A ∪ B depends on the systems A and B only,without any need for the knowledge of the rest of the Universe.Needless to say, A and B are allowed to stay in an arbitrary (a prioriunknown) state.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Groups and Information Theory
G. Sicuro and P.T, preprint (2015).
Problem. How can we put together Information Theory, Einstein’slikelihood principle and Generalized Entropies?A possible merge of these ideas is again offered by the group-theoreticalapproach.
I Main result: All entropies satisfying the first three SK axiomsand the composability axiom do have:
I a) An information-theoretical content.
I b) They satisfy the Einstein’s likelihood principle.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Groups and Information Theory
G. Sicuro and P.T, preprint (2015).
Problem. How can we put together Information Theory, Einstein’slikelihood principle and Generalized Entropies?A possible merge of these ideas is again offered by the group-theoreticalapproach.
I Main result: All entropies satisfying the first three SK axiomsand the composability axiom do have:
I a) An information-theoretical content.
I b) They satisfy the Einstein’s likelihood principle.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Groups and Information Theory
G. Sicuro and P.T, preprint (2015).
Problem. How can we put together Information Theory, Einstein’slikelihood principle and Generalized Entropies?A possible merge of these ideas is again offered by the group-theoreticalapproach.
I Main result: All entropies satisfying the first three SK axiomsand the composability axiom do have:
I a) An information-theoretical content.
I b) They satisfy the Einstein’s likelihood principle.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
The group theoretical information content
Definition 3.7. Assume that S is an entropy satisfying the first three SKaxioms, and that is composable, with
S(A ∪ B) = Φ(S(A),S(B)),
where G is a strictly monotone function such thatΦ(x , y) = G (G−1(x) + G−1(y)). We call the quantity
IG (A) := G−1 (S(A)) . (3)
the information functional associated with the generalized entropy S .Properties.Continuity. IG is continuous respect to its arguments;Maximum principle. IG is maximized on the uniform distribution;Expansibility. The addition of a zero–probability event do not changethe value of I;Additivity. Given two independent systems A and B, in arbitrary states,
IG (A ∪ B) = IG (A) + IG (B). (4)Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Proof of additivity. It is based on the universal property of theLazard formal group: There exists a unique application that mapsthe Lazard group into any other group, in particular into theadditive one. We have:
IG (A ∪ B) = G−1(S(A ∪ B)) = G−1(Φ(S(A),S(B))) (5)
= G−1(G (G−1(S(A) + G−1(S(B))))
= G−1 (S(A)) + G−1 (S(B))
= IG (A) + IG (B),
where A and B are two statistically independent systems, in thestates described by the probability distributions p and q respectively.
I In this context, we shall call G−1(t) the linearization map.I For instance, in the case of Tsallis entropy,
G (t) =e(1−q)t − 1
1− q, G−1(s) =
1
1− qln(1 + (1− q)s)
The linearization map G−1 is exactly the map which transformsTsallis entropy into Reny’s entropy!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Proof of additivity. It is based on the universal property of theLazard formal group: There exists a unique application that mapsthe Lazard group into any other group, in particular into theadditive one. We have:
IG (A ∪ B) = G−1(S(A ∪ B)) = G−1(Φ(S(A),S(B))) (5)
= G−1(G (G−1(S(A) + G−1(S(B))))
= G−1 (S(A)) + G−1 (S(B))
= IG (A) + IG (B),
where A and B are two statistically independent systems, in thestates described by the probability distributions p and q respectively.
I In this context, we shall call G−1(t) the linearization map.
I For instance, in the case of Tsallis entropy,
G (t) =e(1−q)t − 1
1− q, G−1(s) =
1
1− qln(1 + (1− q)s)
The linearization map G−1 is exactly the map which transformsTsallis entropy into Reny’s entropy!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Proof of additivity. It is based on the universal property of theLazard formal group: There exists a unique application that mapsthe Lazard group into any other group, in particular into theadditive one. We have:
IG (A ∪ B) = G−1(S(A ∪ B)) = G−1(Φ(S(A),S(B))) (5)
= G−1(G (G−1(S(A) + G−1(S(B))))
= G−1 (S(A)) + G−1 (S(B))
= IG (A) + IG (B),
where A and B are two statistically independent systems, in thestates described by the probability distributions p and q respectively.
I In this context, we shall call G−1(t) the linearization map.I For instance, in the case of Tsallis entropy,
G (t) =e(1−q)t − 1
1− q, G−1(s) =
1
1− qln(1 + (1− q)s)
The linearization map G−1 is exactly the map which transformsTsallis entropy into Reny’s entropy!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
Einstein’s likelihood function
Definition 3.8. Assume that S is an entropy satisfying the first three SKaxioms, and that is composable, with
S(A ∪ B) = Φ(S(A),S(B)),
where G is a strictly monotone function such thatΦ(x , y) = G (G−1(x) + G−1(y)). The Einstein’s likelihood functionassociated with the entropy S is
W(A) = eG−1(S(A)) (6)
Theorem 3.9.Let S be a composable entropy. Then
W(A ∪ B) =W(A) · W(B).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Proof of Einstein’s principle. We have:
W(A ∪ B) = eG−1(A∪B) = eG−1(Φ(S(A),S(B)))
= eG−1(G(G−1(S(A))+G−1(S(B)))
= eG−1(S(A))+G−1(S(B)) =W(A) · W(B). (7)
I Example. For the Tsallis entropy case, we have the formula:
eG−1(Sq) = eSqq , where eq(x) := [1 + (1− q)x ]
11−q is the
q-exponential. It recovers the formula by Tsallis and Haubold.
I What happens if S is an entropy which is not composable?
I For entropies at least weakly composable, both definitions ofinformation functional IG (A) := G−1 (S(A)) and likekihood
W(A) = eG−1(S(A)) are formally valid, but the additivity andEinstein’s principle are valid only on the uniform distribution. Thisin turn implies that this class of entropies is not very satisfactory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Proof of Einstein’s principle. We have:
W(A ∪ B) = eG−1(A∪B) = eG−1(Φ(S(A),S(B)))
= eG−1(G(G−1(S(A))+G−1(S(B)))
= eG−1(S(A))+G−1(S(B)) =W(A) · W(B). (7)
I Example. For the Tsallis entropy case, we have the formula:
eG−1(Sq) = eSqq , where eq(x) := [1 + (1− q)x ]
11−q is the
q-exponential. It recovers the formula by Tsallis and Haubold.
I What happens if S is an entropy which is not composable?
I For entropies at least weakly composable, both definitions ofinformation functional IG (A) := G−1 (S(A)) and likekihood
W(A) = eG−1(S(A)) are formally valid, but the additivity andEinstein’s principle are valid only on the uniform distribution. Thisin turn implies that this class of entropies is not very satisfactory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Information EntropyRenyi’s entropyEinstein’s likelihood principleGroups and Information Theory
I Proof of Einstein’s principle. We have:
W(A ∪ B) = eG−1(A∪B) = eG−1(Φ(S(A),S(B)))
= eG−1(G(G−1(S(A))+G−1(S(B)))
= eG−1(S(A))+G−1(S(B)) =W(A) · W(B). (7)
I Example. For the Tsallis entropy case, we have the formula:
eG−1(Sq) = eSqq , where eq(x) := [1 + (1− q)x ]
11−q is the
q-exponential. It recovers the formula by Tsallis and Haubold.
I What happens if S is an entropy which is not composable?
I For entropies at least weakly composable, both definitions ofinformation functional IG (A) := G−1 (S(A)) and likekihood
W(A) = eG−1(S(A)) are formally valid, but the additivity andEinstein’s principle are valid only on the uniform distribution. Thisin turn implies that this class of entropies is not very satisfactory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
The composability problem revisitedThe Z-entropies: main definitions
The composability problem revisited
The trace-form class of entropies has a serious drawback:
I Only the Boltzmann and Tsallis entropies are composable.
I This property seems crucial, for thermodynamic andinformation-theoretical purposes.
I The whole plethora of entropies proposed since 1989, is at mostweakly composable. From the previous analysis, this property isnecessary, but probably not sufficient.
I Is there any way out ?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
The composability problem revisitedThe Z-entropies: main definitions
The composability problem revisited
The trace-form class of entropies has a serious drawback:
I Only the Boltzmann and Tsallis entropies are composable.
I This property seems crucial, for thermodynamic andinformation-theoretical purposes.
I The whole plethora of entropies proposed since 1989, is at mostweakly composable. From the previous analysis, this property isnecessary, but probably not sufficient.
I Is there any way out ?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
The composability problem revisitedThe Z-entropies: main definitions
The composability problem revisited
The trace-form class of entropies has a serious drawback:
I Only the Boltzmann and Tsallis entropies are composable.
I This property seems crucial, for thermodynamic andinformation-theoretical purposes.
I The whole plethora of entropies proposed since 1989, is at mostweakly composable. From the previous analysis, this property isnecessary, but probably not sufficient.
I Is there any way out ?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
The composability problem revisitedThe Z-entropies: main definitions
The composability problem revisited
The trace-form class of entropies has a serious drawback:
I Only the Boltzmann and Tsallis entropies are composable.
I This property seems crucial, for thermodynamic andinformation-theoretical purposes.
I The whole plethora of entropies proposed since 1989, is at mostweakly composable. From the previous analysis, this property isnecessary, but probably not sufficient.
I Is there any way out ?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
The composability problem revisitedThe Z-entropies: main definitions
Main result
Main result: There is a new class of non trace-form type of strictlycomposable entropies, coming from formal group theory.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
The composability problem revisitedThe Z-entropies: main definitions
The Z-entropies
P. T., A New Class of Composable Entropies from Group Theory: TheZ-entropies, arxiv: 1507.07436Definition 3.10. Let pii=1,··· ,W , W ≥ 1, with
∑Wi=1 pi = 1, be a
discrete probability distribution. The Z -entropy associated with G isdefined to be the function
ZG ,α(p1, . . . , pW ) :=lnG
(∑Wi=1 pαi
)1− α
, (8)
with 0 < α < 1. Here lnG (x) denotes the generalized group logarithmassociated with G .The Z -entropies are the non trace-form equivalent of the trace-form SU
entropy.They generalized at the same time Boltzmann’s andRenyi’s entropies.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Group logarithms
Let’s come back to the construction of Z -entropies.Definition 3.11. A group logarithm is a continuous, concave andmonotonically increasing function lnG (x) : [1,∞]→ R+ ∪ 0, possiblydepending on a set of real parameters, such that lnG (w) solves thefunctional equation for the group law corresponding to G , i.e
lnG (xy) = Φ(lnG (x), lnG (y)) (9)
where Φ(x , y) = G (G−1(x) + G−1(y)).For instance, when Φ(x , y) = x + y , we have directly that lnG (x) = ln x .If Φ(x , y) = x + y + (1− q)xy , we get for the group logarithm the Tsallislogarithm
lnq =x1−q − 1
q − 1. (10)
Remark 3.12. Any known trace-form entropy provides a grouplogarithm.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Schur-concavity
I Majorization. Given two vectors a, b ∈Rn, we shall say that aweakly majorizes b from below (a w b) if
k∑i=1
a↓i ≥k∑
i=1
b↓i , k = 1, . . . , n,
where a↓i and b↓i are the elements of a and b sorted in decreasingorder. If a w b and also
∑ni=1 ai =
∑ni=1 bi , we say that a
majorizes b (a b).
I An entropy S [p] is said to be Schur-concave if for all probabilitydistributions p = pii∈N and q = qjj∈N such that p ≺ q, we have
S [p] ≥ S [q].
This property is typical of Renyi entropy and is sufficient forguaranteeing that the axiom (SK2) is fulfilled.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Schur-concavity
I Majorization. Given two vectors a, b ∈Rn, we shall say that aweakly majorizes b from below (a w b) if
k∑i=1
a↓i ≥k∑
i=1
b↓i , k = 1, . . . , n,
where a↓i and b↓i are the elements of a and b sorted in decreasingorder. If a w b and also
∑ni=1 ai =
∑ni=1 bi , we say that a
majorizes b (a b).I An entropy S [p] is said to be Schur-concave if for all probability
distributions p = pii∈N and q = qjj∈N such that p ≺ q, we have
S [p] ≥ S [q].
This property is typical of Renyi entropy and is sufficient forguaranteeing that the axiom (SK2) is fulfilled.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I There is a simple criterion to ascertain the Schur-concavity (orconvexity) of a function.Theorem 3.13 (Schur-Ostrowski criterion). Let f be a symmetricfunction, and assume that all first order partial derivatives exist.Then F is Schur-convace if and only if
(xi − xj)
(∂f
∂xi− ∂f
∂xj
)≤ 0.
I Theorem 3.14 (concavity of the Z -family). All the ZG ,α entropiesare concave for 0 < α < 1 and Schur-concave for α > 1.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I There is a simple criterion to ascertain the Schur-concavity (orconvexity) of a function.Theorem 3.13 (Schur-Ostrowski criterion). Let f be a symmetricfunction, and assume that all first order partial derivatives exist.Then F is Schur-convace if and only if
(xi − xj)
(∂f
∂xi− ∂f
∂xj
)≤ 0.
I Theorem 3.14 (concavity of the Z -family). All the ZG ,α entropiesare concave for 0 < α < 1 and Schur-concave for α > 1.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Theorem 3.15. The ZG ,α entropies are all strictly composable, i.e.given any two statistically independent subsystems A, B, defined on anarbitrary probability distribution pii=1,··· ,W , they satisfy thecomposition rule
ZG ,α(A ∪ B) = Ψ(ZG ,α(A),ZG ,α(B)) (11)
where
Ψ(x , y) =1
1− αΦ ((1− α)x , (1− α)y)
with Φ (x , y) being the group law satisfied by the generalized grouplogarithm lnG associated to ZG ,α.
Proposition 3.16. The function Ψ(x , y) is also a group law.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Limiting properties
The class of entropies (1) possesses a simple relation with the most celebratedentropies. In particular, each entropic functional of the class generalizesBoltzmann’s and Renyi’s entropies at the same time, as stated by the followingresults.Proposition 3.17 In the limit α→ 1, the ZG ,α-entropy reduces to theBoltzmann-Gibbs entropy.Proof.
limα→1
lnG
(∑Wi=1 pαi
)1− α = lim
α→1
G[ln(∑W
i=1 pαi
)]1− α =
= limα→1
G ′[ln(∑W
i=1 pαi
)]· 1∑W
i=1 pαi·∑W
i=1 pαi ln pi
−1=
W∑i=1
pi ln1
pi.
where we took into account that G ′(0) = 1.Proposition 3.18 Each of the Z -entropies generalizes the Renyi entropy.Proof. It suffices to observe that the function lnG (x) tends to ln x whenai → 0, i = 1, 2, . . . in the expression of G(t).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Theorem 3.19 The ZG ,α-entropy satisfies the first three SK axioms.(SK1). By definition, the function (8) is continuous with respect to all of itsarguments.(SK2). The entropy (8) is concave for 0 < α < 1. Indeed, given two probability
distributions p(1) =(
p(1)1 , . . . , p
(1)W
)and p(2) =
(p
(21 , . . . , p
(2)W
), we have
ZG ,α(λp1 + (1− λ)p2) =lnG
(∑Wi=1
(λp
(1)i + (1− λ)p
(2)i
)α)1− α ≥
≥lnG
(λ∑W
i=1(p(1)i )α + (1− λ)
∑Wi=1(p
(2)i )α
)1− α ≥
≥ λlnG
(∑Wi=1(p
(1)i )α
)1− α + (1− λ)
lnG
(∑Wi=1(p
(2)i )α
)1− α ,
where we took into account that lnG is by construction monotonically increasingand concave, and that the function
∑Wi=1 pαi is concave for 0 < α < 1.
(SK3). By adding an event of zero probability, we have
ZG ,α(p1, . . . , pW+1) =lnG
(∑W+1i=1 pαi
)1− α =
lnG
(∑Wi=1 pαi
)1− α . (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
A Tower of infinitely many new composable entropies
I 1) Let Φ(x , y) = x + y . We get lnG (x) = ln x . The Renyi entropy
Hα(p1, . . . , pW ) :=ln(∑W
i=1 pαi
)1− α
. (13)
Composition law:
hα(A ∪ B) = Hα(A) + Hα(B)
I 2) Let Φ(x , y) = x + y + qxy . We have lnG (x) = xq−1q .
The associated Zq-entropy is
Zq,α(p1, . . . , pW ) :=
(∑Wi=1 pαi
)q− 1
q(1− α). (14)
Composition law:
Zq,α(A ∪ B) = Zq,α(A) + Zq,α(B) + q(1− α)Zq,α(A)Zq,α(B)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
A Tower of infinitely many new composable entropies
I 1) Let Φ(x , y) = x + y . We get lnG (x) = ln x . The Renyi entropy
Hα(p1, . . . , pW ) :=ln(∑W
i=1 pαi
)1− α
. (13)
Composition law:
hα(A ∪ B) = Hα(A) + Hα(B)
I 2) Let Φ(x , y) = x + y + qxy . We have lnG (x) = xq−1q .
The associated Zq-entropy is
Zq,α(p1, . . . , pW ) :=
(∑Wi=1 pαi
)q− 1
q(1− α). (14)
Composition law:
Zq,α(A ∪ B) = Zq,α(A) + Zq,α(B) + q(1− α)Zq,α(A)Zq,α(B)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Nani gigantum humeris insidentes:
We stay on the shoulders of giants!
(I. Newton).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
A generalization of the Boltzmann, Tsallis, Renyi,Sharma-Mittal entropies
Definition 3.20. The Za,b-entropy is defined to be
Za,b(p1, . . . , pW ) :=
(∑Wi=1 pαi
)a−(∑W
i=1 pαi
)b(a− b)(1− α)
, . (15)
with 0 < α < 1, a < 0, 0 < b < 1.Limiting properties
I Proposition 3.21. The Za,b entropy reduces to the Boltzmannentropy in the limit α→ 1.
I Proposition 3.22. The Za,b entropy reduces to the Renyi entropyin the limit a→ 0, b → 0.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
A generalization of the Boltzmann, Tsallis, Renyi,Sharma-Mittal entropies
Definition 3.20. The Za,b-entropy is defined to be
Za,b(p1, . . . , pW ) :=
(∑Wi=1 pαi
)a−(∑W
i=1 pαi
)b(a− b)(1− α)
, . (15)
with 0 < α < 1, a < 0, 0 < b < 1.Limiting properties
I Proposition 3.21. The Za,b entropy reduces to the Boltzmannentropy in the limit α→ 1.
I Proposition 3.22. The Za,b entropy reduces to the Renyi entropyin the limit a→ 0, b → 0.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I Proposition 3.23. The Za,b entropy reduces to the Tsallis entropyin the limit b → 0, a→ 1.
I Proof.
limb→0
lima→1
(∑Wi=1 pαi
)a−(∑W
i=1 pαi
)b(a− b)(1− α)
= lima→1
(∑Wi=1 pαi
)a− 1
a(1− α)=
=1−
∑Wi=1 pαi
α− 1,
which is the expression of the Tsallis entropy, as a function of theparameter α.
I Proposition 3.24. The Za,b entropy reduces to the Sharma-Mittalentropy in the limit b → 0.Proof. Obvious.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I Proposition 3.23. The Za,b entropy reduces to the Tsallis entropyin the limit b → 0, a→ 1.
I Proof.
limb→0
lima→1
(∑Wi=1 pαi
)a−(∑W
i=1 pαi
)b(a− b)(1− α)
= lima→1
(∑Wi=1 pαi
)a− 1
a(1− α)=
=1−
∑Wi=1 pαi
α− 1,
which is the expression of the Tsallis entropy, as a function of theparameter α.
I Proposition 3.24. The Za,b entropy reduces to the Sharma-Mittalentropy in the limit b → 0.Proof. Obvious.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
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Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I Proposition 3.23. The Za,b entropy reduces to the Tsallis entropyin the limit b → 0, a→ 1.
I Proof.
limb→0
lima→1
(∑Wi=1 pαi
)a−(∑W
i=1 pαi
)b(a− b)(1− α)
= lima→1
(∑Wi=1 pαi
)a− 1
a(1− α)=
=1−
∑Wi=1 pαi
α− 1,
which is the expression of the Tsallis entropy, as a function of theparameter α.
I Proposition 3.24. The Za,b entropy reduces to the Sharma-Mittalentropy in the limit b → 0.Proof. Obvious.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
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Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I The Za,b entropy is related to the Borges-Roditi logarithm:
lnBR(x) :=xa − xb
a− b.
I If we put x = eu (Rota’s isomorphism), we recover what in themathematical literature is called the Abel exponential, given by
expAbel =eau − ebu
a− b=
eγu√δ
sinh(√
δu), (16)
where γ = (a + b)/2,
δ = (a− b)2/4.
I The Abel exponential (16) satisties the Abel functional equation, as wasproven in N. H. Abel, Magazin for Naturvidenskaberne 1 (1823) 216-229.reprinted in: L. Sylow, S. Lie (Eds.), Oeuvres Completes, vol. 1,Christiania, 1881.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I The Za,b entropy is related to the Borges-Roditi logarithm:
lnBR(x) :=xa − xb
a− b.
I If we put x = eu (Rota’s isomorphism), we recover what in themathematical literature is called the Abel exponential, given by
expAbel =eau − ebu
a− b=
eγu√δ
sinh(√
δu), (16)
where γ = (a + b)/2,
δ = (a− b)2/4.
I The Abel exponential (16) satisties the Abel functional equation, as wasproven in N. H. Abel, Magazin for Naturvidenskaberne 1 (1823) 216-229.reprinted in: L. Sylow, S. Lie (Eds.), Oeuvres Completes, vol. 1,Christiania, 1881.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I The Za,b entropy is related to the Borges-Roditi logarithm:
lnBR(x) :=xa − xb
a− b.
I If we put x = eu (Rota’s isomorphism), we recover what in themathematical literature is called the Abel exponential, given by
expAbel =eau − ebu
a− b=
eγu√δ
sinh(√
δu), (16)
where γ = (a + b)/2,
δ = (a− b)2/4.
I The Abel exponential (16) satisties the Abel functional equation, as wasproven in N. H. Abel, Magazin for Naturvidenskaberne 1 (1823) 216-229.reprinted in: L. Sylow, S. Lie (Eds.), Oeuvres Completes, vol. 1,Christiania, 1881.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I The Composition law is the Abel group law:
ΦA(x , y) = x + y + β1xy +∑j>i
βi(
xy i − x iy). (17)
The coefficients βn in (17) can be expressed as
βn =(−1)n−1
n!(n − 1)
∏i+j=n−1, i,j≥0
(ia + jb).
I In particular, if a = −b = k, we have the Z -analog of Kaniadakisentropy:
Zk,α(p1, . . . , pW ) :=
(∑Wi=1 pαi
)k−(∑W
i=1 pαi
)−k
(2k)(1− α). (18)
Zk,α(A ∪ B) = Zk,α(A)√
1 + k2(1− α)2Zk,α(B)2 (19)
+ Zk,α(B)√
1 + k2(1− α)2Zk,α(A)2
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I The Composition law is the Abel group law:
ΦA(x , y) = x + y + β1xy +∑j>i
βi(
xy i − x iy). (17)
The coefficients βn in (17) can be expressed as
βn =(−1)n−1
n!(n − 1)
∏i+j=n−1, i,j≥0
(ia + jb).
I In particular, if a = −b = k, we have the Z -analog of Kaniadakisentropy:
Zk,α(p1, . . . , pW ) :=
(∑Wi=1 pαi
)k−(∑W
i=1 pαi
)−k
(2k)(1− α). (18)
Zk,α(A ∪ B) = Zk,α(A)√
1 + k2(1− α)2Zk,α(B)2 (19)
+ Zk,α(B)√
1 + k2(1− α)2Zk,α(A)2
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I Definition 3.25. Let pii=1,··· ,W , W ≥ 1, with∑W
i=1 pi = 1, be adiscrete probability distribution. Let G be a formal groupexponential, and lnG (x) denotes the associated generalized grouplogarithm (according to Definition 9). The Z δ-entropy associatedwith G is defined to be the function
Z δG ,α(p1, . . . , pW ) :=
lnδG
(∑Wi=1 pαi
)(1− α)δ
. (20)
where 0 < α < 1 if 0 < δ < 1 and α > 1 if δ > 1.
I This class of entropies is concave for 0 < α < 1, 0 < δ < 1 andSchur-concave for α > 1, δ > 1 (according to theSchur-Ostrowski criterion).
I All the class Zδ satisfies the first three SK axiom and in particular isstrictly composable!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I Definition 3.25. Let pii=1,··· ,W , W ≥ 1, with∑W
i=1 pi = 1, be adiscrete probability distribution. Let G be a formal groupexponential, and lnG (x) denotes the associated generalized grouplogarithm (according to Definition 9). The Z δ-entropy associatedwith G is defined to be the function
Z δG ,α(p1, . . . , pW ) :=
lnδG
(∑Wi=1 pαi
)(1− α)δ
. (20)
where 0 < α < 1 if 0 < δ < 1 and α > 1 if δ > 1.
I This class of entropies is concave for 0 < α < 1, 0 < δ < 1 andSchur-concave for α > 1, δ > 1 (according to theSchur-Ostrowski criterion).
I All the class Zδ satisfies the first three SK axiom and in particular isstrictly composable!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
I Definition 3.25. Let pii=1,··· ,W , W ≥ 1, with∑W
i=1 pi = 1, be adiscrete probability distribution. Let G be a formal groupexponential, and lnG (x) denotes the associated generalized grouplogarithm (according to Definition 9). The Z δ-entropy associatedwith G is defined to be the function
Z δG ,α(p1, . . . , pW ) :=
lnδG
(∑Wi=1 pαi
)(1− α)δ
. (20)
where 0 < α < 1 if 0 < δ < 1 and α > 1 if δ > 1.
I This class of entropies is concave for 0 < α < 1, 0 < δ < 1 andSchur-concave for α > 1, δ > 1 (according to theSchur-Ostrowski criterion).
I All the class Zδ satisfies the first three SK axiom and in particular isstrictly composable!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Special case:
Z δG ,α(p1, . . . , pW ) :=
lnδq
(∑Wi=1 pαi
)(1− α)δ
=
(∑Wi=1 pαi
)1−q
− 1
(1− q)(1− α)δ.
This entropy possesses all the good properties of the Tsallis-Cirto entropy(TC) Sq,δ. Precisely, it is extensive when
W (N) = ANµBNν
, A > 0, µ ≥ 0,B > 1, 0 ≤ ν ≤ 1.
In particular, consider the case q = 1. Given a d-dimensional system, theentropy
Z δ(p1, . . . , pW ) :=lnδ(∑W
i=1 pαi
)(1− α)δ
for δ = d/d − 1 is extensive. For d = 3, we have the same beautifulinterpretation already found by Tsallis and Cirto concerning thethermodynamics of black-holes:
Z3/2 ∝ (SBH)3/2.
This entropy does not reduce to the TC entropy; however, it is extensive in thesame regimes, and is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Information-theoretical content: Z-divergences
I Definition 3.26. Let P = pi and R = ri two discrete probabilitydistributions. The Z -divergence associated to G of order α of thedistribution P from the distribution R, denoted by Dα
G , is defined to be
DαG (P||R) :=
lnG
(∑Wi=1
pαirα−1i
)(α− 1)
.
where 0 < α < 1 (here we adopt the standard conventions that 0/0 = 0and x/0 =∞ for x > 0).
I Theorem 3.27 The following inequality holds
DαG (P||R) ≥ 0.
In particular,
DαG (P||R) = 0 if and only if P = R.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Information-theoretical content: Z-divergences
I Definition 3.26. Let P = pi and R = ri two discrete probabilitydistributions. The Z -divergence associated to G of order α of thedistribution P from the distribution R, denoted by Dα
G , is defined to be
DαG (P||R) :=
lnG
(∑Wi=1
pαirα−1i
)(α− 1)
.
where 0 < α < 1 (here we adopt the standard conventions that 0/0 = 0and x/0 =∞ for x > 0).
I Theorem 3.27 The following inequality holds
DαG (P||R) ≥ 0.
In particular,
DαG (P||R) = 0 if and only if P = R.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
Specific example: the Za,b-divergence of order (α, a, b) of the distribution Pfrom the distribution R, denoted by Dα
a,b, is defined to be
Dαa,b (P||R) :=
(∑Wi=1
pαirα−1i
)a
−(∑W
i=1
pαirα−1i
)b
(a− b)(α− 1).
where 0 < α < 1.Observe that
D10,0 (P||R) := lim
α→1Dα
0,0 (P||R) =W∑i=1
pi
(ln
pi
qi
)≥ 0,
i.e. it reduces to the standard Kullback-Leibler divergence. At the same time,
Dα0,0 (P||R) := lim
a→0limb→0
Dαa,b =
1
α− 1ln
(W∑i=1
pαirα−1i
),
which gives the Renyi divergence.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
In short summary:
I Group Law Φ(x , y)←→ Entropy
I Φ(x , y) = G(G−1(x) + G−1(y)⇐⇒ G(t) =∑∞
k=0 aktk+1/(k + 1)
I G(t)←→∑W
i=1 piG(
ln 1pi
)(UGE construction)
I G(t)←→ 11−α lnG
(∑Wi=1 pαi
)(Z-entropy construction)
I where lnG (xy) = Φ(lnG (x), lnG (y))
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
In short summary:
I Group Law Φ(x , y)←→ Entropy
I Φ(x , y) = G(G−1(x) + G−1(y)⇐⇒ G(t) =∑∞
k=0 aktk+1/(k + 1)
I G(t)←→∑W
i=1 piG(
ln 1pi
)(UGE construction)
I G(t)←→ 11−α lnG
(∑Wi=1 pαi
)(Z-entropy construction)
I where lnG (xy) = Φ(lnG (x), lnG (y))
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
In short summary:
I Group Law Φ(x , y)←→ Entropy
I Φ(x , y) = G(G−1(x) + G−1(y)⇐⇒ G(t) =∑∞
k=0 aktk+1/(k + 1)
I G(t)←→∑W
i=1 piG(
ln 1pi
)(UGE construction)
I G(t)←→ 11−α lnG
(∑Wi=1 pαi
)(Z-entropy construction)
I where lnG (xy) = Φ(lnG (x), lnG (y))
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
In short summary:
I Group Law Φ(x , y)←→ Entropy
I Φ(x , y) = G(G−1(x) + G−1(y)⇐⇒ G(t) =∑∞
k=0 aktk+1/(k + 1)
I G(t)←→∑W
i=1 piG(
ln 1pi
)(UGE construction)
I G(t)←→ 11−α lnG
(∑Wi=1 pαi
)(Z-entropy construction)
I where lnG (xy) = Φ(lnG (x), lnG (y))
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Concavity and Schur-concavityLimiting propertiesA Tower of infinitely many new composable entropiesA generalization of the Boltzmann, Tsallis, Renyi, Sharma-Mittal entropiesZ-delta entropiesInformation-theoretical content: Z-divergences
In short summary:
I Group Law Φ(x , y)←→ Entropy
I Φ(x , y) = G(G−1(x) + G−1(y)⇐⇒ G(t) =∑∞
k=0 aktk+1/(k + 1)
I G(t)←→∑W
i=1 piG(
ln 1pi
)(UGE construction)
I G(t)←→ 11−α lnG
(∑Wi=1 pαi
)(Z-entropy construction)
I where lnG (xy) = Φ(lnG (x), lnG (y))
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Conclusions and Future Perspectives
I The Composability Axiom, jointly with the axioms (SK1)-(SK3), allowsto classify the known entropies and to generate infinitely many new ones.
I Under mild hypotheses, a group law Φ(x , y) corresponds to an entropyand vice-versa.
I There are two classes of group entropies.
I The trace-form entropies can be regarded as representation of a uniquegeneral entropy: the universal-group entropy.
I The group entropies of non trace-form type belong to a new family: theZ -class.
I All these entropies are extensive; the trace-form ones are (at least) weaklycomposable; the Z -family is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Conclusions and Future Perspectives
I The Composability Axiom, jointly with the axioms (SK1)-(SK3), allowsto classify the known entropies and to generate infinitely many new ones.
I Under mild hypotheses, a group law Φ(x , y) corresponds to an entropyand vice-versa.
I There are two classes of group entropies.
I The trace-form entropies can be regarded as representation of a uniquegeneral entropy: the universal-group entropy.
I The group entropies of non trace-form type belong to a new family: theZ -class.
I All these entropies are extensive; the trace-form ones are (at least) weaklycomposable; the Z -family is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Conclusions and Future Perspectives
I The Composability Axiom, jointly with the axioms (SK1)-(SK3), allowsto classify the known entropies and to generate infinitely many new ones.
I Under mild hypotheses, a group law Φ(x , y) corresponds to an entropyand vice-versa.
I There are two classes of group entropies.
I The trace-form entropies can be regarded as representation of a uniquegeneral entropy: the universal-group entropy.
I The group entropies of non trace-form type belong to a new family: theZ -class.
I All these entropies are extensive; the trace-form ones are (at least) weaklycomposable; the Z -family is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Conclusions and Future Perspectives
I The Composability Axiom, jointly with the axioms (SK1)-(SK3), allowsto classify the known entropies and to generate infinitely many new ones.
I Under mild hypotheses, a group law Φ(x , y) corresponds to an entropyand vice-versa.
I There are two classes of group entropies.
I The trace-form entropies can be regarded as representation of a uniquegeneral entropy: the universal-group entropy.
I The group entropies of non trace-form type belong to a new family: theZ -class.
I All these entropies are extensive; the trace-form ones are (at least) weaklycomposable; the Z -family is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Conclusions and Future Perspectives
I The Composability Axiom, jointly with the axioms (SK1)-(SK3), allowsto classify the known entropies and to generate infinitely many new ones.
I Under mild hypotheses, a group law Φ(x , y) corresponds to an entropyand vice-versa.
I There are two classes of group entropies.
I The trace-form entropies can be regarded as representation of a uniquegeneral entropy: the universal-group entropy.
I The group entropies of non trace-form type belong to a new family: theZ -class.
I All these entropies are extensive; the trace-form ones are (at least) weaklycomposable; the Z -family is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Conclusions and Future Perspectives
I The Composability Axiom, jointly with the axioms (SK1)-(SK3), allowsto classify the known entropies and to generate infinitely many new ones.
I Under mild hypotheses, a group law Φ(x , y) corresponds to an entropyand vice-versa.
I There are two classes of group entropies.
I The trace-form entropies can be regarded as representation of a uniquegeneral entropy: the universal-group entropy.
I The group entropies of non trace-form type belong to a new family: theZ -class.
I All these entropies are extensive; the trace-form ones are (at least) weaklycomposable; the Z -family is strictly composable.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
I In particular, the Z(a,b) entropy is presented as a new composablegeneralization of many known entropies.
I The group-entropic forms possess many remarkable properties: Legendrestructure, information-theoretical content, etc. (another seminar), allcoming from the group structure.
I A general study of the role of the Z -class in the theory of complexsystems is in order (long-term project).
I The extension of the correspondence among entropies and zetafunctions to the case of multi-parametric entropies and multiple zetavalues and polylogarithms is an open problem.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
I In particular, the Z(a,b) entropy is presented as a new composablegeneralization of many known entropies.
I The group-entropic forms possess many remarkable properties: Legendrestructure, information-theoretical content, etc. (another seminar), allcoming from the group structure.
I A general study of the role of the Z -class in the theory of complexsystems is in order (long-term project).
I The extension of the correspondence among entropies and zetafunctions to the case of multi-parametric entropies and multiple zetavalues and polylogarithms is an open problem.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
I In particular, the Z(a,b) entropy is presented as a new composablegeneralization of many known entropies.
I The group-entropic forms possess many remarkable properties: Legendrestructure, information-theoretical content, etc. (another seminar), allcoming from the group structure.
I A general study of the role of the Z -class in the theory of complexsystems is in order (long-term project).
I The extension of the correspondence among entropies and zetafunctions to the case of multi-parametric entropies and multiple zetavalues and polylogarithms is an open problem.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
I In particular, the Z(a,b) entropy is presented as a new composablegeneralization of many known entropies.
I The group-entropic forms possess many remarkable properties: Legendrestructure, information-theoretical content, etc. (another seminar), allcoming from the group structure.
I A general study of the role of the Z -class in the theory of complexsystems is in order (long-term project).
I The extension of the correspondence among entropies and zetafunctions to the case of multi-parametric entropies and multiple zetavalues and polylogarithms is an open problem.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
Formal groups Z-Entropies The universal-group
entropy
Information Theory
Shannon-Khinchin axioms
and composability
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Information-theoretical aspects of the notion of entropyThe Z-class of Entropies
Main properties of the Z-entropiesState of Art
THANK YOU!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
GROUPS, ENTROPIES AND NUMBERTHEORY
Piergiulio Tempesta
Universidad Complutense de Madridand
Instituto de Ciencias Matematicas (ICMAT),Madrid, Spain.
TEMPLETON SCHOOL ON FOUNDATIONS OF COMPLEXITY
October 14 - 15, 2015
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
LECTURE IV
Formal Groups and Zeta Functions:A Possible Root from Number Theory to
Statistical Mechanics
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
OutlineNumber Theory and Statistical Mechanics
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Preliminaries: The Theory of L-functionsThe Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
Bernoulli numbers and polynomialsSpecial values of zeta functionsFamous CongruencesUniversal structures
Formal Groups and Number TheoryUniversal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Number Theory and Physics
I Perhaps the first connection was established by Montgomery(1970’s): the sequence of zeros of the Riemann zeta function is veryclose to the distributions of energy levels of heavy nuclei!
I Zeta function regularization in quantum field theory (see e.g. theCasimir effect)
I This connection has been extended by considering zeroes ofL-functions and energy levels of heavy nuclei.
I Modern Cryptography over elliptic and hyperelluptic curves
I Polylogarithms, multiple zeta values and Feynman diagrams.
I Are there connections between nonadditive entropies and NumberTheory?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Number Theory and Physics
I Perhaps the first connection was established by Montgomery(1970’s): the sequence of zeros of the Riemann zeta function is veryclose to the distributions of energy levels of heavy nuclei!
I Zeta function regularization in quantum field theory (see e.g. theCasimir effect)
I This connection has been extended by considering zeroes ofL-functions and energy levels of heavy nuclei.
I Modern Cryptography over elliptic and hyperelluptic curves
I Polylogarithms, multiple zeta values and Feynman diagrams.
I Are there connections between nonadditive entropies and NumberTheory?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Number Theory and Physics
I Perhaps the first connection was established by Montgomery(1970’s): the sequence of zeros of the Riemann zeta function is veryclose to the distributions of energy levels of heavy nuclei!
I Zeta function regularization in quantum field theory (see e.g. theCasimir effect)
I This connection has been extended by considering zeroes ofL-functions and energy levels of heavy nuclei.
I Modern Cryptography over elliptic and hyperelluptic curves
I Polylogarithms, multiple zeta values and Feynman diagrams.
I Are there connections between nonadditive entropies and NumberTheory?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Number Theory and Physics
I Perhaps the first connection was established by Montgomery(1970’s): the sequence of zeros of the Riemann zeta function is veryclose to the distributions of energy levels of heavy nuclei!
I Zeta function regularization in quantum field theory (see e.g. theCasimir effect)
I This connection has been extended by considering zeroes ofL-functions and energy levels of heavy nuclei.
I Modern Cryptography over elliptic and hyperelluptic curves
I Polylogarithms, multiple zeta values and Feynman diagrams.
I Are there connections between nonadditive entropies and NumberTheory?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Number Theory and Physics
I Perhaps the first connection was established by Montgomery(1970’s): the sequence of zeros of the Riemann zeta function is veryclose to the distributions of energy levels of heavy nuclei!
I Zeta function regularization in quantum field theory (see e.g. theCasimir effect)
I This connection has been extended by considering zeroes ofL-functions and energy levels of heavy nuclei.
I Modern Cryptography over elliptic and hyperelluptic curves
I Polylogarithms, multiple zeta values and Feynman diagrams.
I Are there connections between nonadditive entropies and NumberTheory?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Number Theory and Physics
I Perhaps the first connection was established by Montgomery(1970’s): the sequence of zeros of the Riemann zeta function is veryclose to the distributions of energy levels of heavy nuclei!
I Zeta function regularization in quantum field theory (see e.g. theCasimir effect)
I This connection has been extended by considering zeroes ofL-functions and energy levels of heavy nuclei.
I Modern Cryptography over elliptic and hyperelluptic curves
I Polylogarithms, multiple zeta values and Feynman diagrams.
I Are there connections between nonadditive entropies and NumberTheory?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Entropies and Number Theory
Main idea. Under suitable regularity hypotheses, we can associate with agroup entropy a class of number-theoretical structures:
Generalized Bernoulli polynomials,
L-series (generalized Riemann-zeta functions),
Hurwitz-type zeta functions,
Polylogarithms, etc.
These generalized objects preserve the same relations possessed by thestandard objects, i.e.:
ζ(s) −→ Bernoulli numbers
Hurwitz zeta function −→ Bernoulli polynomials
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The multiple connections between Number Theory and PhysicsEntropies and Number Theory
Classical Bernoulli
numbers
and polynomials
Riemann zeta function
Hurwitz zeta function
Formal groups
L-functions Generalized Bernoulli
structures
Hyperfunctions
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
The Riemann zeta function
E. C. Titchmarsh The Theory of the Riemann Zeta Function, (1986); H.Iwaniec, E. Kowalski, Analytic Number Theory, AMS, 53 (2004).The Riemann Zeta Function is defined to be the series
ζ (s) =∞∑n=0
1
nss ∈ C, Re s > 1.
It was introduced by Euler for s ∈ R and by Riemann (1860) for s ∈ C. It isabsolutely and uniformly convergent for Re s > 1. It can be represented as aMellin transform:
ζ (s) =1
Γ (s)
∫ ∞0
1
ex − 1x s−1dx .
Euler product:∞∑n=0
1
ns=∏p
1
(1− p−s).
Functional equation:
ξ(s) = ξ(1− s), ξ(s) =1
2π−s/2s(s − 1)Γ
( s2
)ζ(s).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
The Riemann hypothesis: all the non-trivial zeros of the Riemann zetafunction lie in the axis Re s = 1/2: The most important unresolved problemin pure mathematics (Bombieri, 2000).
Figure: ζ(s) doesn’t have any zero on the right of Re s = 1 and on the left ofRe s = 0. Furthermore, the non-trivial zeros are symmetric about the real axisand the line Re s = 1/2. According to the Riemann Hypothesis, they all lie onthe line Re s = 1/2.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
Dirichlet series
T. Apostol, Introduction to Analytic Number Theory, Springer, 1976.A Dirichlet series is a series of the form
L(s) =∞∑n=1
anns, Re s > σ0
These series have great relevance in analytic number theory.Examples.
∞∑n=1
µ(n)
ns=
1
ζ(s),
where µ(n) is the Mobius function, equal to the sum of the primitive n-throots of unity.
∞∑n=1
φ(n)
ns=ζ(s − 1)
ζ(s)
where φ(n) = n∏
p|n (1− 1/p) is Euler’s totient function : it is defined as the
number of integers k ∈ N, 1 ≤ k ≤ n, s.t. gcd(n, k) = 1.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
L-series and L-functions
I Generally speaking an L-series is a Dirichlet series, which has ameromorphic prolongation to the whole complex plane. Thisprolongation defines a complex function, which is the L-functionassociated with the L-series. Usually, one also asks for the existenceof an Euler product.
I In particular, there is a famous axiomatization, due to A. Selberg,of a class of L-series, known as the Selberg class. It is based on fouraxioms (meromorphicity, existence of an Euluer product expansion,existence of a functional equation, Ramanujan’s growth condition.
I We also have other classes: Artin’s L-functions, automorphicL-functions, Dirichlet L-functions, etc.
I A Generalized Riemann hypothesis has been conjectured for theDirichlet L-functions (1884).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
L-series and L-functions
I Generally speaking an L-series is a Dirichlet series, which has ameromorphic prolongation to the whole complex plane. Thisprolongation defines a complex function, which is the L-functionassociated with the L-series. Usually, one also asks for the existenceof an Euler product.
I In particular, there is a famous axiomatization, due to A. Selberg,of a class of L-series, known as the Selberg class. It is based on fouraxioms (meromorphicity, existence of an Euluer product expansion,existence of a functional equation, Ramanujan’s growth condition.
I We also have other classes: Artin’s L-functions, automorphicL-functions, Dirichlet L-functions, etc.
I A Generalized Riemann hypothesis has been conjectured for theDirichlet L-functions (1884).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
L-series and L-functions
I Generally speaking an L-series is a Dirichlet series, which has ameromorphic prolongation to the whole complex plane. Thisprolongation defines a complex function, which is the L-functionassociated with the L-series. Usually, one also asks for the existenceof an Euler product.
I In particular, there is a famous axiomatization, due to A. Selberg,of a class of L-series, known as the Selberg class. It is based on fouraxioms (meromorphicity, existence of an Euluer product expansion,existence of a functional equation, Ramanujan’s growth condition.
I We also have other classes: Artin’s L-functions, automorphicL-functions, Dirichlet L-functions, etc.
I A Generalized Riemann hypothesis has been conjectured for theDirichlet L-functions (1884).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
L-series and L-functions
I Generally speaking an L-series is a Dirichlet series, which has ameromorphic prolongation to the whole complex plane. Thisprolongation defines a complex function, which is the L-functionassociated with the L-series. Usually, one also asks for the existenceof an Euler product.
I In particular, there is a famous axiomatization, due to A. Selberg,of a class of L-series, known as the Selberg class. It is based on fouraxioms (meromorphicity, existence of an Euluer product expansion,existence of a functional equation, Ramanujan’s growth condition.
I We also have other classes: Artin’s L-functions, automorphicL-functions, Dirichlet L-functions, etc.
I A Generalized Riemann hypothesis has been conjectured for theDirichlet L-functions (1884).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
The Hurwitz zeta function
I The Hurwitz zeta function is defined to be the series
ζ(a, s) =∞∑n=1
1
(n + a)s, Re s > 1, a > 0.
This series is absolutely convergent for the given values of s and qand can be extended to a meromorphic function for all s 6= 1. Noticethat ζ(s, 1) = ζ(s).
I Its Mellin transform representation is
ζ (s, a) =∞∑n=0
(n + a)−s =1
Γ (s)
∫ ∞0
e−ax
1− e−xx s−1dx .
I Notice that
ζ(s, a) =1
s − 1
∞∑n=0
(−1)n
n + 1∆n
+a1−s
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
The Hurwitz zeta function
I The Hurwitz zeta function is defined to be the series
ζ(a, s) =∞∑n=1
1
(n + a)s, Re s > 1, a > 0.
This series is absolutely convergent for the given values of s and qand can be extended to a meromorphic function for all s 6= 1. Noticethat ζ(s, 1) = ζ(s).
I Its Mellin transform representation is
ζ (s, a) =∞∑n=0
(n + a)−s =1
Γ (s)
∫ ∞0
e−ax
1− e−xx s−1dx .
I Notice that
ζ(s, a) =1
s − 1
∞∑n=0
(−1)n
n + 1∆n
+a1−s
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
The Hurwitz zeta function
I The Hurwitz zeta function is defined to be the series
ζ(a, s) =∞∑n=1
1
(n + a)s, Re s > 1, a > 0.
This series is absolutely convergent for the given values of s and qand can be extended to a meromorphic function for all s 6= 1. Noticethat ζ(s, 1) = ζ(s).
I Its Mellin transform representation is
ζ (s, a) =∞∑n=0
(n + a)−s =1
Γ (s)
∫ ∞0
e−ax
1− e−xx s−1dx .
I Notice that
ζ(s, a) =1
s − 1
∞∑n=0
(−1)n
n + 1∆n
+a1−s
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
I) NEXT and Hurwitz
I Is there any connection between Nonextensive Statistical Mechanicsand the Hurwitz zeta function?
I Consider the Legendre constraints for the maximization of the Sq
entropy. We know that, under suitable constraints, the canonicaldistributions takes the form
pi = (Zq)−1[1− (1− q)βεi ]1/(1−q).
whereZq =
∑i
[1− (1− q)βεi ]1/(1−q).
I We assume that εk = γ0k, where γ0 is a suitable constant. Then weintroduce
s =1
q − 1, α =
1
β(q − 1)k0We have:
pk =1
(k + α)s/ζ(k, α).
The canonical distribution takes now the form of a Hurwitz distribution.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
I) NEXT and Hurwitz
I Is there any connection between Nonextensive Statistical Mechanicsand the Hurwitz zeta function?
I Consider the Legendre constraints for the maximization of the Sq
entropy. We know that, under suitable constraints, the canonicaldistributions takes the form
pi = (Zq)−1[1− (1− q)βεi ]1/(1−q).
whereZq =
∑i
[1− (1− q)βεi ]1/(1−q).
I We assume that εk = γ0k, where γ0 is a suitable constant. Then weintroduce
s =1
q − 1, α =
1
β(q − 1)k0We have:
pk =1
(k + α)s/ζ(k, α).
The canonical distribution takes now the form of a Hurwitz distribution.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
The Riemann zeta functionL-series and L-functionsThe Hurwitz zeta function
I) NEXT and Hurwitz
I Is there any connection between Nonextensive Statistical Mechanicsand the Hurwitz zeta function?
I Consider the Legendre constraints for the maximization of the Sq
entropy. We know that, under suitable constraints, the canonicaldistributions takes the form
pi = (Zq)−1[1− (1− q)βεi ]1/(1−q).
whereZq =
∑i
[1− (1− q)βεi ]1/(1−q).
I We assume that εk = γ0k, where γ0 is a suitable constant. Then weintroduce
s =1
q − 1, α =
1
β(q − 1)k0We have:
pk =1
(k + α)s/ζ(k, α).
The canonical distribution takes now the form of a Hurwitz distribution.Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The classical Bernoulli numbers and polynomials
I Fermat’s Last Theorem and class field theory (Kummer)
I Theory of Riemann and Hurwitz zeta functions
I Measure theory in p-adic analysis (Mazur)
I Interpolation Theory (Boas and Buck)
I Combinatorics of groups (V. I. Arnold)
I Congruences and Theory of Algebraic Equations
I Ramanujan identities: QFT and Feynman diagrams
I GW invariants, soliton theory (Pandharipande, Veselov)
I More than 1500 papers!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generating functions
Definition. The Bernoulli numbers are the sequence of rational numbersdefined by the generating function
t
et − 1=∞∑k=0
Bk
k!tk .
I The first Bernoulli numbers are
I
B0 = 1, B1 = −1
2,
I
B2 = −1
6, B3 = 0, B4 = − 1
30, B5 = 0,
I
B6 =1
42, B7 = 0, B8 = − 1
30, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generating functions
Definition. The Bernoulli numbers are the sequence of rational numbersdefined by the generating function
t
et − 1=∞∑k=0
Bk
k!tk .
I The first Bernoulli numbers are
I
B0 = 1, B1 = −1
2,
I
B2 = −1
6, B3 = 0, B4 = − 1
30, B5 = 0,
I
B6 =1
42, B7 = 0, B8 = − 1
30, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generating functions
Definition. The Bernoulli numbers are the sequence of rational numbersdefined by the generating function
t
et − 1=∞∑k=0
Bk
k!tk .
I The first Bernoulli numbers are
I
B0 = 1, B1 = −1
2,
I
B2 = −1
6, B3 = 0, B4 = − 1
30, B5 = 0,
I
B6 =1
42, B7 = 0, B8 = − 1
30, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generating functions
Definition. The Bernoulli numbers are the sequence of rational numbersdefined by the generating function
t
et − 1=∞∑k=0
Bk
k!tk .
I The first Bernoulli numbers are
I
B0 = 1, B1 = −1
2,
I
B2 = −1
6, B3 = 0, B4 = − 1
30, B5 = 0,
I
B6 =1
42, B7 = 0, B8 = − 1
30, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
Classical Bernoulli polynomials
Definition. The Bernoulli polynomials are the sequence of polynomialsdefined by the generating function
t
et − 1ext =
∞∑k=0
Bk(x)
k!tk .
I The first Bernoulli polynomials are
I
B0 = 1, B1 = x − 1
2,
I
B2 = x2−x− 1
6, B3 = x3− 3
2x2+
1
2x , B4 = x4−2x3+x2− 1
30,
I
B5 = x5− 5
2x4 +
5
3x3− 1
6x , B6 = x6−3x5 +
5
2x4− 1
2x2 +
1
42, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Classical Bernoulli polynomials
Definition. The Bernoulli polynomials are the sequence of polynomialsdefined by the generating function
t
et − 1ext =
∞∑k=0
Bk(x)
k!tk .
I The first Bernoulli polynomials areI
B0 = 1, B1 = x − 1
2,
I
B2 = x2−x− 1
6, B3 = x3− 3
2x2+
1
2x , B4 = x4−2x3+x2− 1
30,
I
B5 = x5− 5
2x4 +
5
3x3− 1
6x , B6 = x6−3x5 +
5
2x4− 1
2x2 +
1
42, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
Classical Bernoulli polynomials
Definition. The Bernoulli polynomials are the sequence of polynomialsdefined by the generating function
t
et − 1ext =
∞∑k=0
Bk(x)
k!tk .
I The first Bernoulli polynomials areI
B0 = 1, B1 = x − 1
2,
I
B2 = x2−x− 1
6, B3 = x3− 3
2x2+
1
2x , B4 = x4−2x3+x2− 1
30,
I
B5 = x5− 5
2x4 +
5
3x3− 1
6x , B6 = x6−3x5 +
5
2x4− 1
2x2 +
1
42, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
Classical Bernoulli polynomials
Definition. The Bernoulli polynomials are the sequence of polynomialsdefined by the generating function
t
et − 1ext =
∞∑k=0
Bk(x)
k!tk .
I The first Bernoulli polynomials areI
B0 = 1, B1 = x − 1
2,
I
B2 = x2−x− 1
6, B3 = x3− 3
2x2+
1
2x , B4 = x4−2x3+x2− 1
30,
I
B5 = x5− 5
2x4 +
5
3x3− 1
6x , B6 = x6−3x5 +
5
2x4− 1
2x2 +
1
42, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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“Magic” properties of Bernoulli numbers and polynomials
Sums of n-th powers
m∑k=0
kn =Bn+1(m + 1)− Bn+1(0)
n + 1
Appell’s property:∂Bn(x) = nBn−1(x)
Finite differences:∆Bn(x) = nxn−1
Raabe’s theorem (p-adic analysis)
Bn(mx) = mn−1m−1∑k=0
Bn
(x +
k
m
)Fourier series representation:
Bn(x) = − n!
(2πi)n
∑k 6=0
e2πikx
kn
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Special values of zeta functions
There is a close, general connection among special values of zetafunctions and suitable classes of polynomials.
Bernoulli numbers ←→ Riemann’s zeta function
Bernoulli polynomials ←→ Hurwitz’ zeta function
The most basic is the following one:
ζ(1− n) = −Bn
n, n ∈ N
ζ(1− n, a) = −Bn(a)
nn ∈ N, a > 0
Other relations are also considered in the technical literature.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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The Clausen-von Staudt congruence
I The study of congruences in mathematics was started by C. F.Gauss.
I One of the most beautiful congruence of mathematics is that due toClausen and von Staudt, independently. It relates the primenumbers with the Bernoulli numbers.Let p be a prime number, such that p − 1 divides 2k. Then
B2k +∑
p−1|2k
1
p∈ Z
I Prime numbers are both related to the Riemann zeta function andthe Bernoulli numbers
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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The Clausen-von Staudt congruence
I The study of congruences in mathematics was started by C. F.Gauss.
I One of the most beautiful congruence of mathematics is that due toClausen and von Staudt, independently. It relates the primenumbers with the Bernoulli numbers.Let p be a prime number, such that p − 1 divides 2k. Then
B2k +∑
p−1|2k
1
p∈ Z
I Prime numbers are both related to the Riemann zeta function andthe Bernoulli numbers
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
The Clausen-von Staudt congruence
I The study of congruences in mathematics was started by C. F.Gauss.
I One of the most beautiful congruence of mathematics is that due toClausen and von Staudt, independently. It relates the primenumbers with the Bernoulli numbers.Let p be a prime number, such that p − 1 divides 2k. Then
B2k +∑
p−1|2k
1
p∈ Z
I Prime numbers are both related to the Riemann zeta function andthe Bernoulli numbers
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Prime numbers
Formal groups
Special values
Euler product Clausen-von Staudt
congruence
𝜁(𝑠) 𝐵𝑛
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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B2k +∑
p−1|2k
1
p∈ Z
I Example 4.1. B12 = −691/2730. The prime numbers p such thatp − 1 divides 12 are p = 2, 3, 5, 7, 13.
I We have1
2+
1
3+
1
5+
1
7+
1
13− 691
2730= 1 ∈ Z
I Example 4.2. B20 = −174611/330. The prime numbers p suchthat p − 1 divides 20 are p = 2, 3, 5, 11.
I We have1
2+
1
3+
1
5+
1
11− 174611
330= −528 ∈ Z
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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B2k +∑
p−1|2k
1
p∈ Z
I Example 4.1. B12 = −691/2730. The prime numbers p such thatp − 1 divides 12 are p = 2, 3, 5, 7, 13.
I We have1
2+
1
3+
1
5+
1
7+
1
13− 691
2730= 1 ∈ Z
I Example 4.2. B20 = −174611/330. The prime numbers p suchthat p − 1 divides 20 are p = 2, 3, 5, 11.
I We have1
2+
1
3+
1
5+
1
11− 174611
330= −528 ∈ Z
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
B2k +∑
p−1|2k
1
p∈ Z
I Example 4.1. B12 = −691/2730. The prime numbers p such thatp − 1 divides 12 are p = 2, 3, 5, 7, 13.
I We have1
2+
1
3+
1
5+
1
7+
1
13− 691
2730= 1 ∈ Z
I Example 4.2. B20 = −174611/330. The prime numbers p suchthat p − 1 divides 20 are p = 2, 3, 5, 11.
I We have1
2+
1
3+
1
5+
1
11− 174611
330= −528 ∈ Z
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
B2k +∑
p−1|2k
1
p∈ Z
I Example 4.1. B12 = −691/2730. The prime numbers p such thatp − 1 divides 12 are p = 2, 3, 5, 7, 13.
I We have1
2+
1
3+
1
5+
1
7+
1
13− 691
2730= 1 ∈ Z
I Example 4.2. B20 = −174611/330. The prime numbers p suchthat p − 1 divides 20 are p = 2, 3, 5, 11.
I We have1
2+
1
3+
1
5+
1
11− 174611
330= −528 ∈ Z
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
The Almkvist-Meurman congruence
There is a famous congruence for the Bernoulli polynomials, due toAlmkvist and Meurman (1991).Theorem. (Almkvist-Meurman) Let h, k be positive integers. Let
Bn(x) := Bn(x)− Bn(0)
Then
knBn
(h
k
)∈ Z
Example 4.3. Choose B5(x) = x5 − 52x4 + 5
3x3 − x6 . Then
AM(5, h, k) = h5 − 5
2h4k +
5
3h3k2 − hk4
6
Example 4.4. Choose B8(x) = x8 − 4x7 + 143 x6 − 7
3x4 + 23x2 − 1
30 . Then
AM(8, h, k) = h8 − 4h7k +14
3h6k2 − 7
3h4k4 +
2
3h2k6
AM(5, h, k) and AM(8, h, k) are integer numbers for any h, k ∈ N/0.Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
Generalized Bernoulli structure
I Are there generalizations of the Bernoulli numbers and polynomials?
I Surprisingly enough, there are very many: dozens, perhaps hundreds!
I Natural questions:
I A) is there a general structure behind all these generalizations?
I B) In particular, are there fundamental properties which arepreserved?
I C) Are there generalized zeta functions in correspondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generalized Bernoulli structure
I Are there generalizations of the Bernoulli numbers and polynomials?
I Surprisingly enough, there are very many: dozens, perhaps hundreds!
I Natural questions:
I A) is there a general structure behind all these generalizations?
I B) In particular, are there fundamental properties which arepreserved?
I C) Are there generalized zeta functions in correspondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generalized Bernoulli structure
I Are there generalizations of the Bernoulli numbers and polynomials?
I Surprisingly enough, there are very many: dozens, perhaps hundreds!
I Natural questions:
I A) is there a general structure behind all these generalizations?
I B) In particular, are there fundamental properties which arepreserved?
I C) Are there generalized zeta functions in correspondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generalized Bernoulli structure
I Are there generalizations of the Bernoulli numbers and polynomials?
I Surprisingly enough, there are very many: dozens, perhaps hundreds!
I Natural questions:
I A) is there a general structure behind all these generalizations?
I B) In particular, are there fundamental properties which arepreserved?
I C) Are there generalized zeta functions in correspondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generalized Bernoulli structure
I Are there generalizations of the Bernoulli numbers and polynomials?
I Surprisingly enough, there are very many: dozens, perhaps hundreds!
I Natural questions:
I A) is there a general structure behind all these generalizations?
I B) In particular, are there fundamental properties which arepreserved?
I C) Are there generalized zeta functions in correspondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Special values of zeta functionsFamous CongruencesUniversal structures
Generalized Bernoulli structure
I Are there generalizations of the Bernoulli numbers and polynomials?
I Surprisingly enough, there are very many: dozens, perhaps hundreds!
I Natural questions:
I A) is there a general structure behind all these generalizations?
I B) In particular, are there fundamental properties which arepreserved?
I C) Are there generalized zeta functions in correspondence?
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Special values of zeta functionsFamous CongruencesUniversal structures
The Universal Bernoulli Polynomials
P. T., Transactions of the AMS, 367, 7015-7028 (2015). Definition4.1. Let
G (t) =∞∑k=0
aktk+1
k + 1, (1)
be a formal group exponential, where akk∈N a real sequence, witha0 6= 0.The universal higher–order Bernoulli polynomials
B(α)k (x , c1, . . . , cn) ≡ B
(α)k (x) are defined by the relation(
t
G (t)
)αext =
∑k≥0
B(α)k (x)
tk
k!, x , α ∈ R. (2)
Particular cases. If ci = (−1)i , α = 1, then G (t) = et − 1, and theuniversal Bernoulli polynomials and numbers reduce to the standard ones.If ci = (−1)i , α > 1, we get back the higher-order Bernoulli polynomials.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Universal Bernoulli numbers
The quantities Bk ∈ Q [c1, c2, . . .] coincide with Clarke’s universalBernoulli numbers. They satisfy a Universal Clausen–von Staudtcongruence. Let B0 = 1, and if n > 0 is even, then
Bn ≡ −∑p−1|n
p prime
cn/(p−1)p−1
pmod Z [c1, c2, ...] ; (3)
Let B1 = c1/2 and if n > 1 is odd, then
Bn ≡cn1 + cn−3
1 c32
mod Z [c1, c2, ...] . (4)
When cn = (−1)n, the celebrated Clausen–Von Staudt congruence forBernoulli numbers is obtained.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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The universal AM Theorem
Theorem 4.2. Let h > 0, k > 0, n ≥ 0 be integers. Consider thepolynomials defined by
t
G (t)ext =
∑m≥0
BGm (x)
tm
m!,
where
G (t) =∞∑k=0
aktk+1
k + 1, (5)
is a formal group exponential, such that cm ∈ Z for all m. Assume thatcp−1 ≡ 0, 1 mod p for all odd primes p, and either c1 ≡ c3 mod 2, or c1is odd and c3 even. Then
knBGn
(h
k
)∈ Z, (6)
where BGn (x) = BG
n (x)− BGn .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Formal Groups and Number Theory
Universal Bernoulli
Polynomials
and Congruences
Formal groups Integer sequences
Rota’s theory of
Delta Operators
Hyperfunctions
Theory of Functional
Equations
L-series and
L-functions
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Sequences of integer numbers
P. T., Transactions of the AMS, 367, 7015-7028 (2015).Sequences of integer numbers and Appell sequences of polynomials withinteger coefficients are constructed as a byproduct of the previous theory.Lemma 4.3. Consider a sequence of the form
t
G1 (t)− t
G2 (t)=∞∑k=0
Nk
2 k!tk , (7)
where G1 (t) and G2 (t) are formal group exponentials, defined as in formula
(5). Assume that cGj
i ∈ Z for all i ∈ N, j = 1, 2 and cG1p−1 ≡ cG2
p−1 mod p for all
p ≥ 2 (here cGjn denotes the nth coefficient of the expansion for the logarithm
associated to Gj). Then Nkk∈N is a sequence of integers.
Proof. The Bernoulli–type numbers BG1k and BG2
k associated with the formalgroup exponentials G1(t) and G2(t), under the previous assumptions mustsatisfy Clarke’s universal congruence. It follows that the difference BG1
k − BG2k
for k even is an integer and for k odd is a half–integer or an integer.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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a) The characteristic power series obtained as the difference between thepower series associated with the L–genus and that one associated withthe Todd genus, i.e.
Q(t) = t/ tanh t − t/(1− e−t)
is the generating function of the sequence
−1, 1, 0,−1, 0, 3, 0,−17, 0, 155, . . . (8)
b) Realizations of the universal Bernoulli polynomials can be constructedby using delta operators. Here we quote two generating functions ofthese classes of polynomials, related to certain difference delta operators:
∞∑k=0
BVk (x)
k!tk =
text
e3t − 2e2t + 2et − 2e−t + e−2t,
∞∑k=0
BVIIk (x)
k!tk =
−text
e4t − e3t + e2t − 2et + e−t − e−2t + e−3t. (9)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Integer sequences
By combining these delta operators (and different ones), one gets easily apriori infinitely many generating functions of integer sequences.b.1)
t(1 + et)
2(1 + et − e2t).
The sequence generated is 2, 6, 39, 324, 3365, 41958, . . .b.2)
−2t (1 + 2 cosh t + 4 cosh 2t − 6 sinh t)
(−6 + 8 cosh t)(2 + cosh t − cosh 2t − sinh t + sinh 2t + 2 sinh 3t).
The sequence generated is −7, 61,−642, 10127,−207110, 5001663, ...As an immediate consequence of the previous results, we can alsoconstruct new sequences of Appell polynomials possessing integercoefficients.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
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Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Polynomial sequences with integer coefficients
Lemma 4.4Assume the hypotheses of the previous Lemma. Then any sequence ofpolynomials of the form[
t
G1 (t)− t
G2 (t)
]ext =
∞∑k=0
Nk (x)
2k!tk (10)
is an Appell sequence with integer coefficients. As an example, consider thesequence of polynomials pn(x)n∈N, generated by
−te−
3t2 sec h( t
2)[4 + et(1 + 2et(−1 + et))
]2(−3 + 4 cosh t)(1 + (−2 + 4 cosh t) sinh t)
ext . (11)
It is easy to verify that (11) is the generating function of a sequence of Appellpolynomials. The first polynomials of the sequence are
p0(x) = −5, p1(x) = 29− 10x , p2(x) = −150 + 87x − 15x2,
p3(x) = 1279− 600x + 174x2 − 20x3, . . .
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
P. T., L–series and Hurwitz zeta functions associated with the universalformal group, Ann. Scuola Normale Superiore, IX, 1–12 (2010).There is a close correspondence among Formal Groups and AnalyticNumber Theory
I To any formal group law one can associate a family of generalizedBernoulli polynomials:
t
G (t)ext =
∞∑k=0
Bk (x)tk
k!, x ∈ R.
I a generalized Riemann zeta function (L-series) and a generalizedHurwitz zeta function,Let G (t) be a formal group exponential, such that e−at/G (t) is aC∞ function over R+, rapidly decreasing at infinity. The generalizedHurwitz zeta function associated with G is the function ζG (s, a),defined for Re(s) > 1 and a > 0 by
ζG (s, a) =1
Γ (s)
∫ ∞0
e−ax
G (x)x s−1dx . (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
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Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
P. T., L–series and Hurwitz zeta functions associated with the universalformal group, Ann. Scuola Normale Superiore, IX, 1–12 (2010).There is a close correspondence among Formal Groups and AnalyticNumber Theory
I To any formal group law one can associate a family of generalizedBernoulli polynomials:
t
G (t)ext =
∞∑k=0
Bk (x)tk
k!, x ∈ R.
I a generalized Riemann zeta function (L-series) and a generalizedHurwitz zeta function,Let G (t) be a formal group exponential, such that e−at/G (t) is aC∞ function over R+, rapidly decreasing at infinity. The generalizedHurwitz zeta function associated with G is the function ζG (s, a),defined for Re(s) > 1 and a > 0 by
ζG (s, a) =1
Γ (s)
∫ ∞0
e−ax
G (x)x s−1dx . (12)
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
I When a = 0, G (x) = 1− e−x , we obtain the standard Riemannzeta function:
ζ (s) =∞∑n=1
1
ns=
1
Γ (s)
∫ ∞0
1
ex − 1x s−1dx .
I When G (x) = 1− e−x , we obtain the classical Hurwitz zetafunction,
I Theorem 4.5. For any m ∈ N, the following property holds:
ζG (−m, a) = −BG ′
m+1 (a)
m + 1, (13)
where according to eq. (1), BG ′
m (x) is the m–th generalizedBernoulli polynomial associated with the formal group exponentialG ′(t) := −G (−t) (with α = 1).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
I When a = 0, G (x) = 1− e−x , we obtain the standard Riemannzeta function:
ζ (s) =∞∑n=1
1
ns=
1
Γ (s)
∫ ∞0
1
ex − 1x s−1dx .
I When G (x) = 1− e−x , we obtain the classical Hurwitz zetafunction,
I Theorem 4.5. For any m ∈ N, the following property holds:
ζG (−m, a) = −BG ′
m+1 (a)
m + 1, (13)
where according to eq. (1), BG ′
m (x) is the m–th generalizedBernoulli polynomial associated with the formal group exponentialG ′(t) := −G (−t) (with α = 1).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
I When a = 0, G (x) = 1− e−x , we obtain the standard Riemannzeta function:
ζ (s) =∞∑n=1
1
ns=
1
Γ (s)
∫ ∞0
1
ex − 1x s−1dx .
I When G (x) = 1− e−x , we obtain the classical Hurwitz zetafunction,
I Theorem 4.5. For any m ∈ N, the following property holds:
ζG (−m, a) = −BG ′
m+1 (a)
m + 1, (13)
where according to eq. (1), BG ′
m (x) is the m–th generalizedBernoulli polynomial associated with the formal group exponentialG ′(t) := −G (−t) (with α = 1).
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
II) The Tsallis entropy and the Riemann zeta function
The Tsallis entropy is the unique trace-form entropy possessing themultiplicative formal group as the composability law.
This group law is generated by the group exponential G (t) = e(1−q)t−11−q .
Now, if Re s > 1, and q < 1, we have the following result.Theorem 4.6.
1
Γ (s)
∫ ∞0
1e(1−q)t−1
1−q
ts−1dt =1
(1− q)s−1ζ (s) . (14)
In turn, the same group exponential generates the classical Bernoullinumbers and polynomials:
t
et − 1=∞∑k=0
Bktk
k!
text
et − 1=∞∑k=0
Bk(x)tk
k!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Classical Bernoulli
numbers
and polynomials
Riemann zeta function
Tsallis Entropy
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
III) The ζ-entropy
Problem. Can we construct entropies directly related with L-functions?Observe that ζ(x) ∈ A for x ∈ [0, 1).Definition 4.7. The functional
Sζ [p] :=kB
1− aζ ′(a)
W∑i=1
pi
(ζ(apσi )− pσi
σ+
1− ζ(a)
σ
)(15)
with a, σ ∈ (0, 1) will be called the zeta entropy.The entropy Sζ [p] satisfies the first three SK axioms and is weaklycomposable.
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
0.20
0.25
0.30
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
The Ramanujan entropy
The Ramanujan L-function is defined by the relation
L(s) =∞∑n=1
τ(n)
ns, Re s > 6,
where τ(n) is the Ramanujan tau function:∞∑n=1
τ(n)qn = q∏n≥1
(1− qn)24 = η(z)24, q = exp(2πiz), Im z > 0.
This series is absolutely and uniformly convergent, and can be analyticallycontinued to the whole complex plane.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Definition 4.8. The functional
Sτ [p] :=kB
1− aL′(a)
W∑i=1
pi
(L(apσi )− pσi
σ+
1− L(a)
σ
)(16)
will be called the Ramanujan’s τ -entropy. This functional is defined fora, σ ∈ (0, 1).The entropy Sζ [p] satisfies the first three SK axioms and is weaklycomposable.
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
0.20
0.25
0.30
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Resume
We have seen three different connections between Number Theory andNonadditive Entropies.
I NT I) The canonical probability distribution with the Hurwitz zetafunction
I NT II) The relation among generalized logarithms and Dirichletseries (and generalized Hurwitz functions). In particular, Tsallisentropy is directly related to the Riemann zeta function.
I NT III) The existence of nonadditive entropies directly constructedfrom L-functions.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Resume
We have seen three different connections between Number Theory andNonadditive Entropies.
I NT I) The canonical probability distribution with the Hurwitz zetafunction
I NT II) The relation among generalized logarithms and Dirichletseries (and generalized Hurwitz functions). In particular, Tsallisentropy is directly related to the Riemann zeta function.
I NT III) The existence of nonadditive entropies directly constructedfrom L-functions.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Resume
We have seen three different connections between Number Theory andNonadditive Entropies.
I NT I) The canonical probability distribution with the Hurwitz zetafunction
I NT II) The relation among generalized logarithms and Dirichletseries (and generalized Hurwitz functions). In particular, Tsallisentropy is directly related to the Riemann zeta function.
I NT III) The existence of nonadditive entropies directly constructedfrom L-functions.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Conclusions
I C I) The group-theoretical structure coming from thecomposability axiom determines crucially the form and theproperties of generalized entropies, in both cases of trace-formentropies (UGE) and non-trace form ones (Z-entropies).
I C II) In particular, the thermodynamical and information-theoreticalcontent of generalized entropies comes from the group structure.
I C III) The natural language to express the theory is that of formalgroup laws, as was elaborated by Bochner, Novikov, Serre, etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Conclusions
I C I) The group-theoretical structure coming from thecomposability axiom determines crucially the form and theproperties of generalized entropies, in both cases of trace-formentropies (UGE) and non-trace form ones (Z-entropies).
I C II) In particular, the thermodynamical and information-theoreticalcontent of generalized entropies comes from the group structure.
I C III) The natural language to express the theory is that of formalgroup laws, as was elaborated by Bochner, Novikov, Serre, etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Conclusions
I C I) The group-theoretical structure coming from thecomposability axiom determines crucially the form and theproperties of generalized entropies, in both cases of trace-formentropies (UGE) and non-trace form ones (Z-entropies).
I C II) In particular, the thermodynamical and information-theoreticalcontent of generalized entropies comes from the group structure.
I C III) The natural language to express the theory is that of formalgroup laws, as was elaborated by Bochner, Novikov, Serre, etc.
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Conclusions
I C IV) The same group structure is at the heart of manynumber-theoretical constructions: generalized Bernoullipolynomials, integer sequences, a large new class of L-series andL-functions, etc.
I C V) Nonextensive statistical mechanics can be related withL-functions!
I Main result: There are multiple connections among GroupTheory, Generalized Entropies and Analytic Number Theory,still to be explored!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Conclusions
I C IV) The same group structure is at the heart of manynumber-theoretical constructions: generalized Bernoullipolynomials, integer sequences, a large new class of L-series andL-functions, etc.
I C V) Nonextensive statistical mechanics can be related withL-functions!
I Main result: There are multiple connections among GroupTheory, Generalized Entropies and Analytic Number Theory,still to be explored!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Conclusions
I C IV) The same group structure is at the heart of manynumber-theoretical constructions: generalized Bernoullipolynomials, integer sequences, a large new class of L-series andL-functions, etc.
I C V) Nonextensive statistical mechanics can be related withL-functions!
I Main result: There are multiple connections among GroupTheory, Generalized Entropies and Analytic Number Theory,still to be explored!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Trace-form entropies
and
UGE
Universal Bernoulli
Polynomials
and Congruences
Formal groups
L-series
L-functions
Z-Entropies
Information Theory
Rota’s theory of
Delta Operators
Hyperfunctions
Algebraic Topology
Theory of Functional
Equations
Integer sequences
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY
Number Theory and Statistical MechanicsPreliminaries: The Theory of L-functions
Bernoulli numbers and polynomialsFormal Groups and Number Theory
Universal congruences and integer sequencesPolynomial sequences with integer coefficientsFormal Groups and Analytic Number TheoryThe relation between the ζ function and Tsallis entropyThe zeta and Ramanujan entropies
Grazie!!
Piergiulio Tempesta, Universidad Complutense de Madrid and ICMAT GROUPS, ENTROPIES AND NUMBER THEORY