Thermodynamics of abstract composition rules

• Product, addition, logarithm

• Abstract composition rules, entropy formulas and

generalizations of the Boltzmann equation

• Application: Lattice SU2 with fluctuating temperature

T.S.Biró, MTA KFKI RMKI Budapest

Talk given at Zimányi School, Nov. 30. – Dec. 4. 2009, Budapest, Hungary

Thanks to: G.Purcsel, K.Ürmössy, Zs.Schram, P.Ván

Non-extensive Thermodynamics

The goal is to describe:

• statistical

• macro-equilibrium

• irreversible

properties of long-range correlated (entangled) systems

Non-extensive Thermodynamics

The goal is to describe:

• statistical

• macro-equilibrium

• irreversible

properties of long-range correlated (entangled) systems

Non-extensive Thermodynamics

Generalizations done (more or less):

• entropy formulas

• kinetic eq.-s: Boltzmann, Fokker-Planck, Langevin

• composition rules

Most important: fat tail distributions canonically

Applications (fits)

• galaxies, galaxy clusters• anomalous diffusion (Lévy flight)• turbulence, granular matter, viscous fingering• solar neutrinos, cosmic rays• plasma, glass, spin-glass• superfluid He, BE-condenstaion• hadron spectra• liquid crystals, microemulsions• finance models• tomography• lingustics, hydrology, cognitive sciences

Logarithm: Product Sum

additive extensive

Abstract Composition Rules

)y,x(hyx

EPL 84: 56003, 2008

Repeated Composition, large-N

Scaling law for large-N

)0,x(hdy

dx :N

)0,x(hyxx

)0,x(h)y,x(hxx

yy,0x),y,x(hx

2

1n2n1nn

1nn1n1nn

N

1nn0n1nn

Formal Logarithm

Asymptotic rules are associative and attractors among all rules…

Asymptotic rules are associative

).),,((

))()()((

))()(()(

)))()((,()),(,(

1

11

1

zyx

zLyLxLL

zLyLLLxLL

zLyLLxzyx

Associative rules are asymptotic

),(),(

)0(

)(

)0(

)()(

)(

)0(

))0,((

)0()0,(

)()(

)()()),((

0

2

yxhyx

xdz

zxL

xxhxh

yyhh

yxyxh

x

Scaled Formal Logarithm

xxL

axLa

xL

axLa

xL

LL

a

a

)(

)(1

)(

)(1

)(

0)0(,1)0(

0

11

Deformed logarithm

)(ln)/1(ln

))(ln()(ln 1

xx

xLx

aa

aa

Deformed exponential

)()(/1

))(exp()(

xexe

xLxe

aa

aa

Entropy formulas, distributions

Boltzmann – GibbsRényiTsallisKaniadakis …

EPJ A 40: 325, 2009

Entropy formulas from composition rules

Joint probability = marginal prob. * conditional prob.

The last line is for a subset

Entropy formulas from composition rules

Equiprobability: p = 1 / N

Nontrivial composition rule at statistical independence

Entropy formulas from composition rules

ppLp

L

bLaLabL

def

1ln)ln()(

ln

))(())(())((

1

1. Thermodynamical limit: deformed log

Entropy maximum at fixed energy

)()(

)()(

)(

)(

fixed ))()((

max))()((

22

2

2

11

1

1

1221

1

1221

1

ESEX

SYES

EX

SY

EEXEXX

SSYSYY

Generalized kinetic theory

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

Generalized Stoßzahlansatz

)(ln)(ln

)(

0

234123412341

jaiaaijffeG

Fp

pDF

GGwfDF

General H theorem

function rising monotonica

)(G iff 0

))((4

1

)()(

))((

ij

1234341243211234

11

0

jiS

GGw

fDFFS

fFp

pS

General H theorem: entropy density formula

df)f(ln)f(F)f(

)f(ln))f(F(

2)G(ln)G(

a

a

a

Detailed balance: G = G 12 34

Examples for composition rules

Example: Gibbs-Boltzmann

WlnkSW/1ffor

flnfS

)E(eZ

1f

x)x(L

1)0,x(h,yx)y,x(h

eq

2

Example: Rényi, Tsallis

ényi Rln1

1)(

Tsallis )(1

)1(1

),1ln(1

)(

1)0,(,),(

11

/

2

q

nona

aqa

non

a

eqa

fq

SL

ffa

S

aEZ

faxa

xL

axxhaxyyxyxh

Example: Einstein

),(),(

)tanh()(

)tanh(Ar)(

1)0,(

1),(

1

22

2

2

yxhyxc

zczL

c

xcxL

cxxh

cxy

yxyxh

c

c

Important example: product class

axyyxyxa

ezL

axa

xL

axxGxh

xyGyxyxh

az

c

c

),(

1)(

)1ln(1

)(

1)0(1)0,(

)(),(

1

2

Important example: product class

axyyxyxa

ezL

axa

xL

axxGxh

xyGyxyxh

az

c

c

),(

1)(

)1ln(1

)(

1)0(1)0,(

)(),(

1

2

QCD is

like

this

!

Relativistic energy composition

Relativistic energy composition

)cos1(EE2Q

)EE()pp(Q

)Q(UEE)E,E(h

21

2

2

21

2

21

2

2

2121

( high-energy limit: mass ≈ 0 )

Asymptotic rule for m=0

)0(U2/

eq

2

E)0(U21Z

1f

xy)0(U2yx)y,x(

)0(Ux21)0,x(h

Physics background:

q > 1

q < 1

Q²

α

Simulation using non-additive rule

Non-extensive Boltzmann Equation

(NEBE) :

• Rényi-Tsallis energy addition rule

• random momenta accordingly

• pairwise collisions repeated

• momentum distribution collected

with Gábor PurcselPRL 95: 162302, 2005

Evolution in NEBE phase space

Stationary energy distributions in NEBE program

x + y x + y + 2 x y

Thermal equilibration in NEBE program

Scaling variable E or X(E)?Károly Ürmössy

Scaling variable E or X(E)?Károly Ürmössy

Microscopic theory in non-extensive approach: questions, projects, ...

• Ideal gas with deformed exponentials

• Boltzmann and Bose distribution

• Fermi distribution: ptl – hole effect

• Thermal field theory with stohastic temperature

• Lattice SU(2) with Gamma * Metropolis method

As if temperature fluctuated…

• EulerGamma Boltzmann = Tsallis

• EulerGamma Poisson = Negative Binomial

max: 1 – 1/c, mean: 1, spread: 1 / √ c

Euler - Gamma distribution

Tsallis lattice EOS

Tamás S. Bíró (KFKI RMKI Budapest) and

Zsolt Schram (DTP ATOMKI Debrecen)

• Lattice action with

superstatistics

• Ideal gas with power-law tails

• Numerical results on EOS

Lattice theory

A =

DU dt w (t) e t A(U) -S(t,U)c

DU dt w (t) e -S(t,U)c

v

Expectation values of observables:

t = a / a asymmetry parametert s

Action: S(t,U) = a(U) t + b(U) / t

Su2 Yang-Mills eos on the lattice with Euler-Gamma distributed inverse temperature: Effective action method

preliminary

with Zsolt Schram (work in progress)

Method: EulerGamma * Metropolis

• asymmetry thrown from Euler-Gamma

• at each Monte Carlo step / only after a while

• at each link update / only for the whole lattice

• meaning local / global fluctuation in space

• c = 1024 for checking usual su2

• c = 5.5 for genuine quark matter

Ratio

e / T4

(e-3p) / T4

Ideal Tsallis-Bose gas

For c = 5.5 we have 1 / a = 4.5 and e ≈ 4 e_0

Summary

• Non-extensive thermodynamics is not only

derivable from composition rules, but it is

realized by QCD interactions in the high-

energy limit and can be seen in heavy-ion

collisions!

Topical Review Issue of EPJ A