The thermodynamical limit of abstract composition rules

T. S. Bíró, KFKI RMKI Budapest, H

• Non-extensive thermodynamics

• Composition rules and formal log-s

• Repeated rules are associative

• Examples

Talk by T. S. Biro at Varoš Rab, Dalmatia, Croatia, Sept. 1. 2008.

From physics to composition rules

Entropy is not a sum: correlations in the common probability

Energy is not a sum: (long range) interaction inside the system

Thermodynamical limit: extensive but not additive?

Short / long range interaction vs. extensivity

dN

dN

dconst

N

E

rrrg

rdrrgnEN

E

d

N

dN

/1

1

ln

.

)()(

)()(

short range

long range

Short / long range correlation vs. extensivity

dN

dN

dconst

N

S

rrgrg

rdrgrgnSN

S

d

N

dN

/1

1

ln

.

)(ln)(

)(ln)(

short range

long range

Typical g( r ) functions

ga

scrystal strin

gy

liqu

id

From physics to composition rules

Abstract composition rule h(x,y)

anomalous diffusion

multiplicative noise

coupled stochastic equations superstatistics

fractal phase space filling

chaotic dynamics

power-law tailed distributions extended logarithm and exponential

Lévy distributions

From composition rules to physics

Abstract composition rule h(x,y)

h(x,0) = x, general rules

associative (commutative) rules

Formal logarithm L(x)

equilibrium distribution: exp ּס L

generalized entropy: L̄ ¹ ּס ln

Thermodynamical limit:

repeated rules

Thermodynamical limit: repeated rules

N-fold composition

Thermodynamical limit: repeated rules

recursion

00x

Thermodynamical limit: repeated rules

use the ‘ zero property ’ h(x,0) = x

Thermodynamical limit: repeated rules

The N limit: scaling differential equation

Thermodynamical limit: repeated rules

solution: asymptotic formal logarithm

Note: t / t_f = n / N finite ratio of infinite system sizes (parts’ numbers)

Thermodynamical limit: repeated rules

The asymptotic rule is given by

Proof of associativity:

Thermodynamical limit: associative rules are attractors

If we began with h(x,y) associative, then it has a formal logarithm, F(x).

Proportional formal logarithm same composition rule!

Boltzmann algorithm: pairwise combination + separation

With additive composition rule at independence:

Such rules generate exponential distribution

Boltzmann algorithm: pairwise combination + separation

With associative composition rule at independence:

Such rules generate ‘exponential of the formal logarithm’ distribution

Entropy formulae from canonical equilibrium

Equilibrium: q – exponential, entropy: q - logarithm

All composition rules generate a non-extensive entropy formula in the th. limit

Entropy formulae from canonical equilibrium

Dual views: either additive or physical quantities

Associative composition rules can be viewed as a canonical equilibrium

Rules and entropies

h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln

composition rule formal logarithmformal

exponentialequilibrium distribution

entropy formula

ffLS

eeZf

ttL

xxL

yxyxh

f

EEL

eq

ln)(ln

)(

)(

),(

11

)(

1

Gibbs, Boltzmann

Rules and entropies

h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln

composition rule formal logarithmformal

exponentialequilibrium distribution

entropy formula

a

ffLS

aEeZf

etL

axxL

axyyxyxh

a

f

aEL

eq

ata

a

1

11

/)(

11

1

)(ln

1

1)(

1ln)(

),(

Pareto, Tsallis

Rules and entropies

h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln

composition rule formal logarithmformal

exponentialequilibrium distribution

entropy formula

a

f

EEL

eq

a

a

aaa

fLS

eeZf

ttL

xxL

yxyxh

a

/111

)(

/11

/1

ln)(ln

)(

)(

),(

Lévy

Rules and entropies

h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln

composition rule formal logarithmformal

exponentialequilibrium distribution

entropy formula

c

c

f

ff

c

cvcvvL

eq

xcxcc

cLS

eZf

ctctL

xL

cxy

yxyxh

/2

/2

1

111

2/

/1/1)(

1

2

2

)(ln

)/tanh()(

ln)(

1),(

Einstein

Classification based on h’(x,0)

• constant addition Gibbs distribution

• linear Tsallis rule Pareto

• pure quadratic Einstein rule rapidity

• quadratic combined Einstein-Tsallis

• polynomial multinomial rule rational

function of power laws

Interaction and kinematics

• Assume that the interaction energy can be

expressed via the asymptotic, free individual

energies. This gives an energy composition

law as:

),(212112EEUEEE

Interaction and kinematics

Let U depend on the relative momentum

squared:

)cos(2

)cos(2

)(

2

21

2

21

2

2

1112

baQ

ppmEEQ

QUEEE

Interaction and kinematics

Average over the directions gives for the kinetic

energy composition rule (with F’ = U)

2/122/1221

2121

21

41

2112

2111

)(

)0()2()2(

)22()22(

Km

Km

b

KKb

KKKKma

UmKUmKU

baFbaFKKK

Interaction and kinematics

In the extreme relativistic limit (K >> m) it gives

)0(4

)0()4(

21

21

2112U

KK

FKKFKKK

Rule and asymptotic rule

h(x,y) L(x) L̄ ¹(t) exp ּס L L̄ ¹ ּס ln

Original composition rule

Asymptotic formal logarithm

formal exponential

equilibrium distribution

entropy formula

1)(ln

1

1)(

)0(1ln)(

0)0()(),(

111

/)(

11

1

aaf

aEL

eq

ata

a

fLS

aEeZf

etL

GaaxxL

GxyGyxyxh

Pareto - Tsallis

Interaction and kinematics

In the non-relativistic limit (K << m) the angle

averaged composition rule has the form:

)()()0(1)0,(

)0()()(

4

)2()2(),(

32

2xUxUUxh

UyUxU

xy

xyyxFxyyxFyxyxh

x

non-trivial formal logarithm non-additive entropy formula

Summary

• Extensive composition rules in the

thermodynamical limit are associative and

symmetric, they define a formal logarithm, L

• The stationary distribution is exp o L, the

entropy is related to L inverse o ln.

• Extreme relativistic kinematics leads to the

Pareto-Tsallis distribution.