Review of Tsallis distribution applied to RHIC data?

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

pDF

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