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:
• Tsallis and similar:
• Extensive S(E,V,N)
• non-extensive
• 0:
• 0:
an absolute
temperature exists
???
• 1: energy is conserved
• 1: (quasi) energy is
conserved
• 2: entropy does not
• 2: entropy does not
decrease spontan.
decrease
NEBE parton cascade
Boltzmann equation:

f1 
t
w
1234
 f3 f4
 f1 f 2

234
w 1234  M
2
1234
   p1  p 2  p 3  p 4
  h( E 1 , E 2 )  h( E 3 , E 4 ) 
Special case: E=|p|

Energy composition rule
Associative rule  mapping to addition: quasi-energy
h ( h ( x , y ), z )  h ( x , h ( y , z ))
X ( h)  X ( x )  X ( y )
h( x , y )  x  y 
Taylor expansion for small x,y and h
X  ( 0 )
X(0)
xy  
Stationary distribution in NEBE
Gibbs of the additive quasi-energy = Tsallis of energy
Boltzmann-Gibbs in X(E)
1 X ( E )/T
f ( p) 
e
Z
Generic rule
h ( x , y )  x  y  a xy
Quasi-energy
Tsallis distribution
X(E) 
1
a
ln 1  aE 
1
1  aE   1 / aT
f ( p) 
Z
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
S ( E 1 , E 2 )  max .
h ( E 1 , E 2 )  const .
S 1 dE 1  S 2 dE 2  0
h1 dE 1  h 2 dE 2  0
S 1 ( E 1 , E 2 ) h1 ( E 1 , E 2 )

S 2 ( E 1 , E 2 ) h 2 ( E 1 , E 2 )
 ( E1 )   ( E 2 )  
If LHS = RHS thermal equilibrium, if same function: universal temperature
Thermodynamics: normal case
S ( E 1 )  S ( E 2 )  max .
E 1  E 2  const .
S  ( E 1 ) dE 1  S  ( E 2 ) dE 2  0
dE 1  dE 2  0
S ( E1 )
1

S ( E 2 )
1
S ( E1 )  S ( E 2 )  
If LHS = RHS thermal equilibrium, if same function: universal temperature
Thermodynamics: NEBE case
S ( E 1 )  S ( E 2 )  max .
S  ( E 1 ) dE 1  S  ( E 2 ) dE 2  0
X ( E 1 )  X ( E 2 )  const .
X  ( E 1 ) dE 1  X  ( E 2 ) dE 2  0
S ( E1 )
X ( E1 )

S ( E 2 )
X ( E 2 )
S ( E1 )
S ( E 2 )

 
X ( E1 )
X ( E 2 )
If LHS = RHS thermal equilibrium, if same function: universal temperature
Thermodynamics: Tsallis case
Tsallis entropy: S(E1,E2) = S1 + S2 + (q-1) S1 • S2;  Y(S) additiv, Rényi
Y ( S ( E 1 ))  Y ( S ( E 2 ))  max .
E 1  E 2  const .
Y  ( S ( E 1 )) S  ( E 1 ) dE 1  Y  ( S ( E 2 )) S  ( E 2 ) dE 2  0
dE 1  dE 2  0
Y  ( S ( E 1 ))  S  ( E 1 )
1


Y ( S ( E 2 ))  S ( E 2 )
Y  ( S ( E 1 ))  S  ( E 1 )  Y  ( S ( E 2 ))  S  ( E 2 )  
If LHS = RHS thermal equilibrium, if same function: universal temperature
Thermodynamics: NEBE case
S   X(E) 

f ln f   f X ( E p )  max
1  X ( E p )
f 
e
Z
X ( E )
1

 
ln f ( E ) 
 S ( E )
T slope ( E )
E
T
T slope
T

 T ( 1  aE )  T  ( q  1 ) E
X ( E )
 = 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=0.12 GeV,
q=1.2388
T=0.08 GeV,
q=1.1648
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)