equilibration of non-extensive systems
DESCRIPTION
Equilibration of non-extensive systems. NEBE parton cascade Zeroth law for non-extensive rules Common distribution Extracting temperatures. T. S. Bíró and G. Purcsel MTA KFKI RMKI Budapest. Talk given at Varos Rab, Croatia, Aug.31-Sept.3 2007. Boltzmann – Gibbs: Extensive S(E,V,N) - PowerPoint PPT PresentationTRANSCRIPT
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)