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 , 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 EPL 84: 56003, 2008

Repeated Composition, large-N

Scaling law for large-N

Formal Logarithm

Asymptotic rules are associative and attractors among all rules…

Asymptotic rules are associative

Associative rules are asymptotic

Scaled Formal Logarithm

Deformed logarithm Deformed exponential

Formal composition rules Differentiable rules Asymptotic rules Associative rules Formal Logarithm 1.General rules repeated infinitely  asymptotic rules 2.Asymptotic rules are associative 3.Associative rules are self-asymptotic 4.For all associative rules there is a formal logarithm mapping it onto the simple addition 5.It can be obtained by scaling the general rule applied for small amounts

Entropy formulas, distributions Boltzmann – Gibbs Rényi Tsallis Kaniadakis … 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 1. Thermodynamical limit: deformed log

Entropy maximum at fixed energy

Canonical distribution The fit value of T = 1 / β correlates to that of the non-extensivity parameter ‘a’

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

General H theorem

General H theorem: entropy density formula

Detailed balance: G = G 12 34

Detailed balance: G = G 12 34

Detailed balance: proof

Examples for composition rules

Example: Gibbs-Boltzmann

Example: Rényi, Tsallis

Example: Einstein

Example: Non associative

Important example: product class

QCD is like this!

Relativistic energy composition

( high-energy limit: mass ≈ 0 )

Asymptotic rule for m=0

Physics background: q > 1 q < 1 Q²Q² 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: , 2005

Evolution in NEBE phase space

Stationary energy distributions in NEBE program x + yx + y + 2 x y

Thermal equilibration in NEBE program

Non-extensive spectra: quark and hadron level Check quark coalescence scaling 2:3 Assume that T,q stem from quark matter Assume that hadronization is rapid Assume a transverse blast wave Result: Quark number scaling for (q-1) with. Károly Ürmössy EPJ A 40:325,2009

Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing

Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra SQM 2008, Beijing

Blast wave fits and quark coalescence SQM 2008, Beijing

Non-extensive spectra: hadron level Scaling variable for p_T dependence T,q for each particle  universality? Assume a transverse blast wave (for high-p_T it is ok) Result: Hadronic T,q and v parameters with. Károly Ürmössy work in progress

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

Limiting temperature with Tsallis distribution Hagedorn Massless particles, 3-dim. momenta, N-fold For N  4: Tsallis partons  Hagedorn states ( with A. Peshier, Giessen ) PLB 632:247,2006 If it were a ~ 1/T  there would not be a limiting T !!!

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 parameter ts 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