Equilibration of non-extensive systems

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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
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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

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NEBE parton cascade
Boltzmann equation
Special case Ep
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Energy composition rule
Associative rule ? mapping to addition
quasi-energy
Taylor expansion for small x,y and h
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Stationary distribution in NEBE
Gibbs of the additive quasi-energy Tsallis of
energy
Boltzmann-Gibbs in X(E) Generic
rule Quasi-energy Tsallis distribution
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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

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Boltzmann energy equilibration
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Tsallis energy equilibration
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Boltzmann distribution equilibration
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Tsallis distribution equilibration
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Mixed distribution equilibration
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Mixed distribution equilibration
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Thermodynamics general case
If LHS RHS thermal equilibrium, if same
function universal temperature
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Thermodynamics normal case
If LHS RHS thermal equilibrium, if same
function universal temperature
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Thermodynamics NEBE case
If LHS RHS thermal equilibrium, if same
function universal temperature
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Thermodynamics Tsallis case
Tsallis entropy S(E1,E2) S1 S2 (q-1) S1
S2 ? Y(S) additiv, Rényi
If LHS RHS thermal equilibrium, if same
function universal temperature
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Thermodynamics NEBE case
? 1 / T in NEBE the inverse log. slope is
linear in the energy
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Boltzmann temperature equilibration
T 0.50 GeV
T 0.32 GeV
T 0.14 GeV
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Tsallis temperature equilibration
T0.16 GeV, q1.3054
T0.12 GeV, q1.2388
T0.08 GeV, q1.1648
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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)