Post on 23-Jan-2016
description
Review of Tsallis distribution applied to RHIC data?
NO!Theory about why is it applicable: yes!
T.S.Bíró, G.Purcsel and K.Ürmössy
MTA KFKI RMKI Budapest
Talk given at Zimányi Winter School, 2008. nov. 25-29. Budapest, Hungary
Transverse momentum spectra
TT
aT
vpmE
aEZ
f
/11
1
Transverse momentum spectra
TT
aT
vpmE
aEZ
f
/111
Transverse momentum spectra
TT
aT
vpmE
aEZ
f
/111
What all it can…• dN/dy rapidity distribution: reduced phase
space (q < 1)
• Multiplicity distribution: negative binomial
• Temperature – average energy fluctuation
• Superstatistics
• Coalescence scaling for (q-1)
Theory: thermodynamics with power-law
tailed energy distributions
Power-law tailed distributions and abstract composition rules
•Extensivity and non-extensivity
•Composition rules in the large-N limit
• Entropy formulas and distributions
• Relativistic kinetic energy composition
T.S.Bíró, MTA KFKI RMKI Budapest
Talk given at Zimanyi Winter School, 2008. nov. 25-29. Budapest, Hungary
Extensivity and non-extensivity
T.S.Biro, arxiv:0809.4675 Europhysics Letters 2008
T.S.Biro, G.Purcsel, Phys.Lett.A 372, 1174, 2008
T.S.Biro, K.Urmossy, G.G.Barnafoldi, J.Phys.G
35:044012, 2008
Extensive is not always additive
2112
...12lim
xxx
N
xN
N
Nonextensive as composite sums
),(
))((
2112
1
1
1
11...12
lim
limlim
xxhx
x
xLLN
x
N
iiN
N
N
iiN
N
N
N
Pl. x_i=i, L(x)=exp(ax)
Composition, large-N limit
),( yxhyx
Composition, large-N limit
),( :rule Asymptotic
)( :yExtensivit
),()( :Composed
)0,( :propertyImportant
2121
limN
1
NNNN
N
Ny
Ny
N
N
xxx
yx
hhyx
xxh
Composition, large-N limit
)0,( :N Large
),(
)0,(),(
),( :Recursion
2
2121
111
1
xhydt
dx
xhxx
xhxhxx
xhx
Ny
nNy
nn
nNy
nnn
Ny
nn
n = t N
Composition by formal logarithm
Asymptotic rules are associative
).),,((
))()()((
))()(()(
)))()((,()),(,(
1
11
1
zyx
zLyLxLL
zLyLLLxLL
zLyLLxzyx
Associative rules are asymptotic
),(),(
)0(
)(
)0(
)()(
)(
)0(
))0,((
)0()0,(
)()(
)()()),((
0
2
yxhyx
xdz
zxL
xxhxh
yyhh
yxyxh
x
Associative rules are
attractors
among more general rules
Entropy formulas, distributions
Formal logarithm
xxL
axLa
xL
axLa
xL
LL
a
a
)(
)(1
)(
)(1
)(
0)0(,1)0(
0
11
Deformed logarithm
)(ln)/1(ln
))(ln()(ln 1
xx
xLx
aa
aa
Deformed exponential
)()(/1
))(exp()(
xexe
xLxe
aa
aa
Non-extensive entropy and energy
)(
)ln(1
anon
anon
LfE
fLfS
Entropy maximum at fixed energy
)()(
)()(
)(
)(
fixed ))()((
max))()((
22
2
2
11
1
1
1221
1
1221
1
ESEX
SYES
EX
SY
EEXEXX
SSYSYY
Canonical distribution and
detailed balance solution in
generalized Boltzmann equation:
T
EXefaeq
)(
Example: Gibbs-Boltzmann
ffS
EeZ
f
xxL
xhyxyxh
eq
ln
)(1
)(
1)0,(,),(2
Example: Tsallis
ényi Rln1
1)(
Tsallis )(1
)1(1
),1ln(1
)(
1)0,(,),(
11
/
2
q
nona
aqa
non
a
eqa
fq
SL
ffa
S
aEZ
faxa
xL
axxhaxyyxyxh
Example: Kaniadakis
)(2
1
)1(1
)sinh(Ar1
)(
1)0,(
11),(
11
/22
22
2
2222
ffS
xxZ
f
xxL
xxh
xyyxyxh
non
eq
Example: Einstein
),(),(
)tanh()(
)tanh(Ar)(
1)0,(
1),(
1
22
2
2
yxhyxc
zczL
c
xcxL
cxxh
cxy
yxyxh
c
c
Example: Non associative
yxyx
zazLa
xxL
axh
yx
xyayxyxh
c
c
),(
)1()(1
)(
1)0,(
),(
1
2
Example: Non associative
axyyxyxa
ezL
axa
xL
axxGxh
xyGyxyxh
az
c
c
),(
1)(
)1ln(1
)(
1)0(1)0,(
)(),(
1
2
Generalized Boltzmann equation
)(ln)(ln
)(
0
234123412341
jaiaaijffeG
Fp
GGwfDF
H theorem
function rising monotonica
)(G iff 0
))((4
1
)()(
))((
ij
1234341243211234
11
0
jiS
GGw
fDFFS
fFp
pS
H theorem
dfffFf
ffF
GG
a
a
a
)(ln)()(
)(ln))((
)(ln)(
Relativistic energy composition
Relativistic energy composition
B
BAFBAFQU
BAQ
EEppQ
QUEEEEh
4
)22()22()(
)cos(2
)()(
)(),(
2
2
2
21
2
21
2
2
2121
Angle averaged Q dependent composition rule for the
relativistic kinetic energies
)2)(2(44
2/)()(
))0,0(2()),0(2())0,(2(
)!12(
)4()2(),(
21
2
2
2112
0
2
)2(
mymxxyB
mmymxmxyA
AUyAUxAU
j
BAUyxyxh
j
j
j
Formal logarithm and asymptotic rule for the relativistic kinetic
energies
2
212
1234
122
)(2
)()2(
)()(2)0(21)0,(
mmxmz
zUxmxm
zUxmUmxh
Asymptotic rule for m=0
)0(2/
eq
2
)0(211
)0(2),(
)0(21)0,(
UEUZ
f
xyUyxyx
Uxxh
Summary
• Non-extensive thermodynamics requires
non-additive composition rules
• Large N limit: rules are associative and
symmetric, formal logarithm L
• Entropy formulas, equilibrium
distributions, H-theorem based on L
• Relativistic kinematics Tsallis-Pareto
Appendix: how to throw new momenta?
qdhEEdw
PqqPmE
qPp
pdpdhEEPpp
3
21
22412
2,1
2
2,1
21
2,1
2
3
1
3
2121
cos
,
Appendix: how to throw new momenta?
ddEdEP
EEhEEdw
ddqPqdEdEEE
dPqdqPqdEE
dddqqqd
21
21
21
2
2121
21
21
2,12,1
23
sin
sincos
sin
Appendix: how to throw new momenta for BG?
)6,0( interval thein
rnd uniform 32
1
)(
3
3
1
2
1
1
11
Ph
EhE
P
ddEP
EhEdw
Appendix: how to throw new momenta for Tsallis?
),0( interval thein rnd uniform
1
)2(1ln)2(
1
)1(
)(
max
1
11
13
12
1
11
aE
aEaEaEah
Pa
ddEaEP
EhEdw