Equilibration of non-extensive systems

T. S. Bíró and G. Purcsel

MTA KFKI RMKI Budapest

• NEBE parton cascade

• Zeroth law for non-extensive rules

• Common distribution

• Extracting temperatures

Talk given at Varos Rab, Croatia, Aug.31-Sept.3 2007

Thermodynamics

• Boltzmann – Gibbs:

• Extensive S(E,V,N)

• 0: an absolute

temperature exists

• 1: energy is conserved

• 2: entropy does not

decrease spontan.

• Tsallis and similar:

• non-extensive

• 0: ???

• 1: (quasi) energy is

conserved

• 2: entropy does not

decrease

NEBE parton cascade

Boltzmann equation:

Special case: E=|p|

Energy composition rule

Associative rule mapping to addition: quasi-energy

Taylor expansion for small x,y and h

Stationary distribution in NEBE

Gibbs of the additive quasi-energy = Tsallis of energy

Boltzmann-Gibbs in X(E)

Generic rule

Quasi-energy

Tsallis distribution

Abilities of NEBE

• Tsallis distribution from any initial distribution

• Extensiv (Boltzmann-) entropy

• Particle collisions in 1, 2 or 3 dimensions

• Arbitrary free dispersion relation

• Pairing (hadronization) option

• Subsystem indexing

• Conserved N, X( E ) and P

Boltzmann: energy equilibration

Tsallis: energy equilibration

Boltzmann: distribution equilibration

Tsallis: distribution equilibration

Mixed: distribution equilibration

Mixed: distribution equilibration

Thermodynamics: general case

If LHS = RHS thermal equilibrium, if same function: universal temperature

Thermodynamics: normal case

If LHS = RHS thermal equilibrium, if same function: universal temperature

Thermodynamics: NEBE case

If LHS = RHS thermal equilibrium, if same function: universal temperature

Thermodynamics: Tsallis case

If LHS = RHS thermal equilibrium, if same function: universal temperature

Tsallis entropy: S(E1,E2) = S1 + S2 + (q-1) S1 • S2; Y(S) additiv, Rényi

Thermodynamics: NEBE case

= 1 / T in NEBE; the inverse log. slope is linear in the energy

Boltzmann: temperature equilibration

T = 0.50 GeV

T = 0.32 GeV

T = 0.14 GeV

Tsallis: temperature equilibration

T=0.16 GeV, q=1.3054

T=0.08 GeV, q=1.1648

T=0.12 GeV, q=1.2388

Summary

• NEBE equilibrates non-extensive

subsystems

• It is thermodynamically consistent

• There exists a universal temperature

• Not universal but equilibrates: different T

and a systems (not different T and q

systems: Nauenberg)