1. 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

2. 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

3. NEBE parton cascade Boltzmann equation: Special case: E=|p|

4. Energy composition rule Associative rule  mapping to addition: quasi-energy Taylor expansion for small x,y and h

5. Stationary distribution in NEBE Gibbs of the additive quasi-energy = Tsallis of energy Boltzmann-Gibbs in X(E) Generic rule Quasi-energy Tsallis distribution

6. 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

7. Boltzmann: energy equilibration

8. Tsallis: energy equilibration

9. Boltzmann: distribution equilibration

10. Tsallis: distribution equilibration

11. Mixed: distribution equilibration

13. Thermodynamics: general case If LHS = RHS thermal equilibrium, if same function: universal temperature

14. Thermodynamics: normal case If LHS = RHS thermal equilibrium, if same function: universal temperature

15. Thermodynamics: NEBE case If LHS = RHS thermal equilibrium, if same function: universal temperature

16. 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

17. Thermodynamics: NEBE case  = 1 / T in NEBE; the inverse log. slope is linear in the energy

18. Boltzmann: temperature equilibration T = 0.50 GeV T = 0.32 GeV T = 0.14 GeV

19. Tsallis: temperature equilibration T=0.16 GeV, q=1.3054 T=0.08 GeV, q=1.1648 T=0.12 GeV, q=1.2388

20. 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)