Dissipative effects on relativistic hydrodynamics

in the presence of long range interactions

based on arXiv:0710.1905[nucl-th]

T.O and G. Wilk PRC77 044903, (2008).

T. Osada （ Tokyo City Univ. ） and G. Wilk （ Andrzej Sołtan Inst. ）

mini-Workshop at Dept. of Phys. TCU. June 24, 2009

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9RHIC　Experiments

Lengthof thering 3.8km

accelerated Au upto

0.99999

and collide head-on

v c

The detector of PHENIX　 Collab.

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日経サイエンス　２００６年０８月号

初期宇宙における高温・高密度状態の再現・解明に向けて高エネルギー原子核衝突実験⇔極限状態での物質の形態

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Non- extensivity for system with correlations

hydL

lcorrelation and/or fluctuation scale;of composing particles

quarks, gluonsor

hadronsproduced byhigh-energy

nucleus-nucleuscollisions

hydro scale; definable local equilibrium or stationary

T

1

correlation

fluctuationq ~ l /Lhyd entropy S state

Lhyd ≥ l ≥ ↦ 0

q = 1 extensiveBoltzmann-Gibbs

equilibrium

0 ≤ Lhyd ＜ l

q >1Nonextensive

stationary( ) ( ) ( )S A B S A S B

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Assumptions in Boltzmann-Gibbs statistics

some assumptions leading to the Boltzmann-Gibbs (BG) statistics 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

2

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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

l1

1n

q

qi

i

p

qp

( ) ( ) (1 ) ( ) )( ) (q qq q qS A B S A S B q S A S B

Tsallis’s non-extensive entropy

B lnqqq i i

iS k p p

3

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Why Nonextensive hydrodynamics?

* ‘standard’ thermodynamics exponential particle spectra, experiments definitely power-law tail @ high pT.

↦ usual hydro + (other dynamical origins…) or non-hydrodynamic approach

↦ Nonextensive hydro + (other dynamical origins…)

↑ including (momentum) correlation

* Nonextensive (perfect) hydro ⇌ (usual q=1) dissipative hydrolink via nonextensive/dissipative correspondence

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Relativistic non-extensive kinetic theory

non-extensive version of Boltzmann equation:

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

33 31 11

0 0 01 11 1

1

1

( , ) ( , ),

( , | ,[ , ]

[ , ]

)1( , )

( , | , )2q 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 p

W p p p pp p h fp

f

f

1 1binary collisions: p p p p

1 1

1 1 1 Boltzmann Stosszahlansatz - (for =1)

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

[ , ] ( , ) ( , )q q

q q

h ff h f x p f x p

h ff 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

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Non-extensive H theorem

1 1

1/ (1 )

[ , ] exp [ln ln ]

exp ( ) 1 (1 )

q q

q

q

q

q qh

X q X

ff ff

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

3

B 3 0

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

( ) ( , )ln ( , )(

))

,2

(qq q q

qq qf

d p ps x k f x p f

ppx p x

q-generalized Boltzmann Stosszahlansatz

3

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

qq q q q q

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

p

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 f

p

k d pf C x p

p

revised by Osada and Wilk therm. dyn. rel. OK

q-generalized entropy current

6

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q- equilibrium

3

3 0

1( ) ( , )

(2 )

q

q q

d pT x p p f x p

p

q- energy-momentum tensor:

3

0[ ] ( , ) ( , ) 0, if ( , ) ( ) ( )q

d pF x p C x p x p a x b x p

p

B3( ) [ln ] 0

(2 )q q q

ks x F f

q- equilibrium distribution function:

11

B

( )( , ) 1 (1 )

( )

q

q

p u xf x p q

k T x

collision invariant:

B

( )setting ( ) 0, ( )

( )

u xa x b x

k T x

setting ( ) 0, ( ) const.a x b x

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q- hydrodynamical model

1+1 ( - ) relativistic -hydrodynamics:q

1( ) 0

1( ) 0

q q q q qq q q

q q q q qq q q

vP v

v P PP v

1( ) cosh( ), sinh( ) , ( ) tanh( )q q q q qu x v x

2 22

1 1 metric =(1,- ), , ln

2t z

g t zt z

; ;( ) 0q q q q q qT P u u P g

8

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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 p

p

( ) ( ) (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 p

s x k f x p f x p f x pp

( ) ( ) ( )S A B S A S B

standard hydrodynamics, locally conserve (BG) entropy current:

without correlations between cells

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hydL hydL hydLhydL

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Single particle spectra by q-hydro model 10

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Single particle spectra by q-hydro model 11

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

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

2

2

1[ 2 ]; ,

3

[1 ] ,[1 ]

q q q q q

q q q q qq

w w P u u

W WW w u w u u

w

; ;[ ( ) ( ( ) ) ] 0q T u u P T WT u W u

( )

( ) ( ) ( ) ( ) 3q

q q q q

if exist T and u x satisfying

P T P T and T T

dissipative hydrodynamics

(perfect) nonextensive hydrodynamics

Nonextensive/dissipative correspondence 12

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Entropy production

;;

usu T

T

13q > 1 ↦ near equilibrium; stationary stateq = 1 ↦ true equilibrium

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Bulk and shear viscosities

1

full;2 2

1 3 ( 2)/ (1/ 4 ) 3( 1)

w Zsu X

s s T

perfect q-hydrodynamics + nonextensive/dissipative correspondence

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Summary

Nonextensive (Tsallis’s ) 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 the next ? 2+1 q-hydro model with QGP EoS.

↦ elliptic flow (pT-dependence) and the HBT puzzle q-energy momentum tensor in Einstein equation ↦ effects on the cosmological constant

15