高エネルギー重イオン衝突のための相対論的Tsallis...

高エネルギー重イオン衝突のための

相対論的Tsallis流体模型の研究

武蔵工大知識工学部長田剛

Andrzej Sołtan Inst. G. Wilk

基礎物理学研究所研究会熱場の量子論とその応用京都大学 2007.0９.０６

基研研究会熱場の量子論とその応用９月６日

Scales in hydrodynamical model

macroscopic scalethe system’s size

L

microscopic scale(e.g. mean free path)

lideal hydrodynamics

- strongly coupled system -l L

‘cell’ of the hydrodynamics


Standard hydrodynamics based on extensivityThe basic assumption of standard statistical mechanics is that the system under consideration can be subdivided into a set of non-overlapping subsystems.

B ln ,

( ) ( ) ( )

i ii

S k p p

S A B S A S B

= − ∑

⊕ = +

As a consequence the Boltzmann-Gibbs entropy is extensive in the sense that the total entropy of two independent subsystems is the sum of their entropies.

extensivity


Assumptions in Boltzmann-Gibbs factor

some assumptions leading tothe Boltzmann-Gibbs (BG) factor may be too tight?

* absence of memory effects,* negligible local correlation* absence of long-range interaction

+ Boltzmann H-theorem (based on the extensive entropy)

extensivity:entropy, measure of information about the particle distribution in the states available to the system, is extensive in the sense that the total entropy of two independent subsystems is the sum of their entropies.

B ln ,

( ) ( ) ( )

i ii

S k p p

S A B S A S B

= − ∑

⊕ = +


Correlations and non- extensivityIf memory effects and long-range forces are present, this property is no longer valid and the entropy, which is a measure of the information about the particle distribution in the states available to the system, is not an extensive quantity.

B ln ,

( ) ( ) ( )

i ii

S k p p

S A B S A S B

= − ∑

⊕ ≠ +non- extensivity

★ memory effect (temporal correlation) ★ long range forces (spatial correlation)

are present


.

Non-extensive entropy

・In 1988 Tsallis proposed a generalization of the entropy of the BG entropy C. Tsallis, J.Stat.Phys.52(1988) 749.

1

B

1ln

1

l ,nq

q

q i iq

iq

ff

q

S k p p− −

≡−

= − ∑

( ) ( ) (1 ) ( ) )( ) (q qq q qS A B S A S B q S A S B+ −⊕ = +

Tsallis’s non-extensive entropy


Introduction: Why Tsallis- hydro?

• the equilibrium in ‘standard’ thermodynamics relies on exponential particle spectra, while experiments definitely show a power-law tailin high energy transverse momenta.

⇒ hydro +(other dynamical origins…) ≟ Tsallis- hydro + (other dynamical origins…)

↑including (momentum) correlation

• ideal Tsallis (q- )hydro = (usual q=1) dissipative hydro ?

q: a parameterof non-extensivityintroduced by C. Tsallis in 1988.

q=1: usual the Boltzmann-Gibbs thermodynamics,statistics.


Relativistic non-extensive kinetic theorynon-extensive version of Boltzmann equation:

A. Lavagno, Phys.Lett.A301(2002) 13.

33 31 11

0 0 01 11 1

1

1

( , ) ( , ),

( , | ,[ , ]

[ , ]

)1( , )( , | , )2

q q q

q q q

qq q

q

p f x p C x p

W p p p pd pd p h fd pC x pW p p p pp p h fp

ff

µµ

′

∂ =

′ ′⎧ ⎫′′ ⎪ ⎪= ⎨ ⎬′ ′′ ′ ⎪⎩

′

− ⎪⎭∫

1 1binary collisions: p p p p′ ′+ → +

1 1

1 1 1 Boltzmann Stosszahlansatz - (for =1)

[ , ] [ ( , ), ( , )]

[ , ] ( , ) ( , )q q

q q

h f f h f x p f x ph f f f x p f x p→ −

′ ′=

′ ′= ⋅

★ correlation function same space-time x but different p

1 1 1 1

transition rate between two particle state, assuming the detailed-balance: ( , | , ) ( , | , )W p p p p W p p p p•

′ ′ ′ ′=


Non-extensive H theorem

q-generalized entropy current

[ ]1 1

1/(1 )

[ , ] exp [ln ln ]

exp ( ) 1 (1 )

q q

q

q

q

q qh

X q X

f f f f−= + −

= +

J. A. S. Lima, R. Silva and A.R. Plastino,Phys.Rev.Lett86(2001),2983

{ }3

B 3 0( ) ( , )ln ( , ) ( , )(2 )

qq q q q q

d p ps x k f x p f x p f x pp

µµ

π≡ − −∫

{ }3

B 3 0

by A .Lavagno,Phys.Lett.A301 (2002 )13 .

( ) ( , ) ln ( , )(

))

,2

(qq q q

qq qfd p ps x k f x p f

ppx p x

µµ

π≡ − −∫

3B

3 0

3B

3 0

( ) [ln ](2 )

[ln ] ( , ) 0 for all space-time point(2 )

qq q q q

q q q

k d ps x f p fp

k d p f C x pp

µ µµ µπ

π

∂ = − ∂

= − = ≥

∫

∫

revised by Osada and Wilktherm. dyn. rel. OK

q-generalized Boltzmann Stosszahlansatz


q- equilibrium

3

3 0

setting ( ) 0, ( ) const.

1( ) ( , )(2 )

qq q

a x b x

d pT x p p f x pp

µ

µν µ ν

π= =

= ∫

q- energy-momentum tensor:

3

0 [ ] ( , ) ( , ) 0

collision invaria

if

nt:

( , ) ( ) ( )

qd pF x p C x pp

x p a x b x p µµ

ψ ψ

ψ

= ≡

= +

∫

B3

B

( )( ) [ln ] 0 setting ( ) 0, ( )

(2 ) ( )q q q

u xks x F f a x b xk T x

µµµ µπ

−∂ = = = = −

q- equilibrium distribution function:1

1

B

( )( , ) 1 (1 )

( )

q

q

p u xf x p q

k T x

µµ

−⎡ ⎤= − −⎢ ⎥

⎢ ⎥⎣ ⎦


Thermodynamic relations

42 2 2 /(1 )

2

42 2 1/(1 )

2

32 2 /(1 ) 1/(1 )

2

[1 (1 ) ]2

[1 (1 ) ]2

[1 (1 ) ] [1 (1 ) ]2

q qq z

qq z

q q qq z

gT de e z e q e

gTP de e z e q e

gTs de e z e q e q e

επ

π

π

−

−

− −

= − − −

= − − −

⎡ ⎤= − − − + − −⎣ ⎦

∫

∫

∫

(Gibbs-Duhem)

q q q

qq

Ts PdP

sdT

ε= +

=

/z m T=

hold also for the q- generalized quantities.


q- hydrodynamical model

1+1 ( - ) relativistic -hydrodynamics:qτ η

1( ) 0

1( ) 0

q q q q qq q q

q q q q qq q q

vP v

v P PP v

ε ε α αε

τ τ η τ τ η

α αε

τ τ η τ τ η

∂ ∂ ∂ ∂⎧ ⎫+ + + + =⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭

∂ ∂ ∂ ∂⎧ ⎫+ + + + =⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭

1( ) cosh( ), sinh( ) , ( ) tanh( )q q q q qu x v xµ α η α η α ητ

⎡ ⎤≡ − − ≡ −⎢ ⎥⎣ ⎦

2 22

1 1 metric =(1,- ), , ln2

t zg t zt z

µν τ ητ

+≡ − ≡

−

; ;( ) 0q q q q q qT P u u P gµν µ ν µν

µ µε⎡ ⎤= + − =⎣ ⎦


Standard hydrodynamics vs. q- hydrodynamics

Tsallis hydrodynamics,locally conserve q- entropy current:

{ }3

B 3 0( ) ( , )ln ( , ) ( , )(2 )

qq q q q q

d p ps x k f x p f x p f x pp

µµ

π≡ − −∫

( ) ( ) (1 ) ( ) )( ) (q qq q qS A B S A S B q S A S B+ −⊕ = +

including correlations

{ }3

B 3 0( ) ( , )ln ( , ) ( , )(2 )d p ps x k f x p f x p f x p

p

µµ

π≡ − −∫

( ) ( ) ( )S A B S A S B⊕ = +

standard hydrodynamics, locally conserve entropy current:

without correlations between cells

much smaller than system size- strongly coupled system -

much smaller than system size- strongly coupled system -


1E-4 1E-3 0.01 0.1 1 10

0.20

0.25

0.30

0.35P q /

ε q

εq [GeV/fm3]

q =1.0 q =1.1 q =1.2

q- EoS: non-extensive relativistic π gas

- 0/ , 0.14GeV, 3( for , , )z m T m g π π π+= = =

42 2 2 /(1 )

2

42 2 1/(1 )

2

[1 (1 ) ]2

[1 (1 ) ]2

q qq z

qq z

gT de e z e q e

gTP de e z e q e

επ

π

−

−

= − − −

= − − −

∫

∫


q- Gaussian Initial condition

2(in)

0 2

0

( 1fm, ) exp ,2

( 1fm, )

q q

q

ηε τ η εσ

α τ η η

⎡ ⎤= = −⎢ ⎥

⎣ ⎦= =

BRAHMS Collab.Phys.Rev.Lett.93(2004),102301

2tot T 0 0 0 T

tot part part

( , ), 1.0fm, 6.5fm

, 357, 73 6GeV qE A d A

E N E N E

π τ η ε τ η τ= = =

= ∆ = ∆ = ±∫

RHIC Au + Au 200GeV/nucleon


q- initial condition (for fixed σ )

0 1 2 3 40

5

10

15

20

25

30

q =1.00, ε(in)q =28.7 GeV/fm3

q =1.05, ε(in)q =25.9 GeV/fm3

q =1.10, ε(in)q =22.4 GeV/fm3

q =1.15, ε(in)q =16.8 GeV/fm3

Initial condition: σ = 1.25 fixed

η

ε q( τ 0, η

)


Single particle spectra by q- hydro model

0 1 2 3 4 5 610-5

10-4

10-3

10-2

10-1

100

101

102

103

1/2π

1/p T

dN

/dp T

dy

pT [GeV/c]

q=1.08 q=1.09 q=1.10STAR Collab.

Au+Au 200GeV/nucleon

σq=1.30, ε(in)=20.5 [GeV/fm3]

TF=100 MeV


Single particle spectra by q- hydro model

0 1 2 3 4 5 60

50

100

150

200

250

300

350

BRAHMS Au+Au 200GeV/nucleon

dN/d

y

y0 2 4 6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

q=1.08TF=100 MeV

STAR Au+Au 200GeV/nucleon

σq=1.26 ε(in)=22.6 [GeV/fm3]

σq=1.28 ε(in)=21.5 [GeV/fm3]

σq=1.30 ε(in)=20.5 [GeV/fm3]

σq=1.32 ε(in)=19.5 [GeV/fm3]

σq=1.34 ε(in)=18.5 [GeV/fm3]

σq=1.36 ε(in)=17.6 [GeV/fm3]

1/2π

1/p T

dN

/dp T

dy

pT [GeV/c]


Perfect q-hydrodynamics = (q=1) dissipative hydrodynamics?

;

;

[ ]

[ ( ) ] 0q q q q q qT u u P

u u P W u W u

µν µ ν µνµ

µ ν µν µ ν ν µ µνµ

ε

ε π

= + ∆

= + + Π ∆ + + + =

rewusi ritng ,P and , ,

in a dissipative like form

e

:qu Tformallyµ µνε

q q

q q

q q

P P P P

u u u uµ µ µ µ

ε ε ε ε→ ≡ + ∆

→ ≡ + ∆

→ ≡ + ∆

non-extensivity q affects thermodynamical quantities and flow velocity

,

1 ( ) ,2( )

q q

q q q q

q q q q

P P P

W u P u

P u u

µ µ µ

µν µ ν

ε ε

π ε

Π ≡ ∆ = −

≡ ∆ + + ∆

≡ + ∆ ∆


q≠1 near equilibrium picture

1 stationary state near ( 1)equilibriumq

q≠

=

eq eq 1,q qT T T T T

T W u W u

µν µν µν µν µν

µν µν µ ν ν µ µν

δ

δ π== + ≡

= Π∆ + + +

;;entropy production: usu T

Tµ µνν

µµδ⎡ ⎤ = −⎣ ⎦

Landau matching condition [U. Heinz, et.al., Phys.Rev.C73,034904(2006) ]

0u T uµνµ νδ = guarantee to use the same of ( ) as of ( ).

qT T T Tε ε

solution exsists only for 1 caseu qµ <

1 'ture' equilibriumq ≠

q- hydrodynamical model:


Entropy production in the q≠1 near equilibrium picture

0 2 4 6 810-6

10-5

10-4

10-3

10-2

10-1

100

[suµ ]

;µ

σ=1.25 fixed

q=0.95

τ=5.0 fm τ=25.0 fm τ=75.0 fm

ηηη0 2 4 6 8

0 2 4 6 8

;; usu T

Tµ µνν

µµδ⎡ ⎤ = −⎣ ⎦


Summary

Tsallis- (nonextensive-) hydrodynamical model is formulated by based on the relativistic nonextensive kinetic theory.perfect q- hydrodynamics may be connected with the dissipative ‘standard’( i.e., q=1) hydrodynamics.

What’s next ?2+1 or full dimension q-hydro model with QGP EoS.write down the fluctuation-dissipation theorem

for a nonextensive version and find transport-coefficients using q.


Appendix:non-extensive and assumptions some assumptions leading to

the Boltzmann-Gibbs (BG) factor may be too tight?* absence of memory effects,

interactions can be described as a successions of simple binary collisionsalways possible determine a time interval does not change appreciably and its rate of change at time t depends only on its instantaneous value and not on its previous history.

* negligible local correlationassumptions for the space dependence of the distribution function its rate of change at a spatial point depends only on the neighborhood of that point.i.e., the range of the interactions is short with respect to the characteristic spatial dimension of the system.

* absence of long-range interactionthe momenta of two particles at the same spatial point are not correlated and the corresponding two-body correlation function can be factorized as a product of two single particle distributions.

+ Boltzmann H-theorem (based on the extensive entropy)


q- hydrodynamical evolution of Temperature T

3in 20GeV/fmε =

0 2 4 6 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8σ=1.25 fixed

q = 1.00

η

0 2 4 6 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 τ =1fm τ =5fm τ =25fm τ =75fm

q = 1.10

T [G

eV]

η


q- flow vs. q- energy density

10-2 10-1 100 1010.0

0.1

0.2

0.3

0.4

0.5σ=1.25 fixed

q = 1.00

αq-

η

10-2 10-1 100 101

0.1

0.2

0.3

0.4

0.5

εq [GeV/fm3] εq [GeV/fm3]

τ = 1.2 fm τ = 1.4 fm τ = 1.6 fm τ = 1.8 fm τ = 2.0 fm

q = 1.10


Single particle spectra: freeze out surface

f

3

3 3 ( , )Cooper-Frye: (2 ) q

d N gE d fpp

pd

xµµσ

π Σ= ∫

0 2 4 60

50

100

150

200

250

300

TF=100 MeV

τ [f

m]

η

σ=1.25 fixed ε(in)=28.7 [GeV/fm3] fixed

0 2 4 60

50

100

150

200

250

300

q = 1.00 q = 1.05 q = 1.10 q = 1.15

η

高エネルギー重イオン衝突のための相対論的Tsallis...

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高エネルギー重イオン衝突のための 相対論的Tsallis...

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