Dynamics of the phase transition in nuclear matter

Stahlberg D.S.

Dubna International Advanced School of Theoretical Physics,Helmholtz International Summer School, July 14-26, 2008

in collaboration with Skokov V.V. and Toneev V.D.

Contents

IntroductionThe relaxation approximation for phase transition in nuclear matterApplication to Bjorken expansion with phase transitionApplication to 1+1 D hydrodynamic model in the Cartesian coordinatesSummary

Introduction

Hydrodynamics as applied to heavy-ion collisions gives us possibilities for

extracting information about global properties of compressed and hot nuclear

matter. In this talk I consider a hydrodynamic approach to description of evolution

of system produced in heavy-ion central collisions.

Nuclear matter may be in different phase states and suffers a phase transition.

http://www.gsi.de/

How can this phenomena influence

on hydrodynamics of expanding

fireball ?

The main goal of our research is to

investigate the first order phase

transitions in mixed phase region

and apply obtained knowledge to

hydrodynamic description of system.

In the first steps of the research we use the simple model of first order phase

transition in nuclear matter with zero chemical potential. This model let us get

to know, what we should expect in more complicated cases.

It based on the developed 1+1 dimensional hydrodynamic model with

dynamics of phase transition. Assuming an existence of hadron, quark-gluon

and mixed phases in nuclear matter we describe the phase transition taking

into account thermal nonequilibrium in the system.

The relaxation approximation for phase transition in nuclear matter

.0

,0

=∂

=∂μ

μ

μνν

BN

T

In perfect fluid approximation

.,)(

μμ

μννμμν ε

unNPguuPT

BB =

−+=

Conservation laws:

If both phases have the same collective velocity , we can use following expressions for each of them:

μu

,

,)(

μμ

μννμμν ε

unN

gPuuPT

iBiB

iiii

=

−+=

where defines the phase.IIIi ,=

,)1( μνμνμν λλ III TTT −+=;)1(

μμμ λλ IIBIBB NNN −+=

Mixed phase is a mixture of hadron and quark-gluon phases.

VVI /=λ

(1)

We assume the existence of mechanical equilibrium for mixed phase )( PPP III ==

and consider the system with zero chemical potentials . )0( == IIBIB μμ

Then we have PguuPT IIIμννμμν ελλε −+−+= ))1((

III

Rewrite equation (1) as a set of equations, taking into account (2) and (3) :

(2)

ελλεε )1( −+= (3)

.0)(

,0)(

=∂−∂+∂+

=∂++∂

PPuuuuP

uPu

νμμ

ννμμ

ννν

ν

ε

εε

To simulate thermal non-equilibrium we define in relaxation time approximation:λ

. λλλ −=∂Γ eqt

Here is given by the Gibbs conditions: eqλ .,, IIIIIIIII TTPP μμ ===(5)

(4)

(6)

If the relaxation time then 0=Γ eqλλ = and phase transition proceeds in a way

defined by the Gibbs conditions (6). (Maxwell construction)

Obviously in mixed phase changes in the range from 0 to 1λ

Hadron EoS:

.3

.3dd

.3,90

,

4

24

II

III

III

II

P

TPTPT

ggTP

ε

αε

παα ππ

=

=−=

===

.34

3

.3dd

.GeV/fm 1946.0

.16,90

,

4

3

24

BP

BTPTPT

B

ggBTP

IIII

IIIIII

IIIIII

ggIIII

−=

+=−=

=

==−=

ε

βε

πββ

Quark-gluon EoS:

Gibbs conditions (6) define the temperature of phase transition

GeV. 18.04 ≈−

=αβ

BTpt

.GeV/fm 9131.03)(

,GeV/fm 1347.03)(

3

3

≈+−

=

≈−

=

BBT

BT

ptII

ptI

αββε

αβαε

.4

)()(

BTptII

eq

εεελ

−=

We solve system (4) completed by (7) or (8) in case of pure hadron or quark-gluon phase, respectively.

(7) (8)

In the mixed phase case we solve system (4) completed by relaxation equation (5) and expression (9) for .eq

(9)

λ

Application to Bjorken expansion with phase transition

1=Dassuming .

Fig.1: Energy density vs. the proper time for various values of the relaxation time Γ

Fig.2: Pressure dependence on the proper time for various relaxation times

.dd

,0)(dd

λλτλ

εττ

ε

−=Γ

=++

eq

PD

Γ

Initial conditions:

30

0

GeV/fm 2.75)(

fm/c1

=

=

τε

τ

accords to )( 0τεGeV A158=labE( )

(10)

Fig. 3: Pressure as a function of energy density for various relaxation times . (Effective EoS)Black triangles illustrate the proper time steps.The value of step is equal to 1 fm/c.

Fig. 4: Ratio P/ as a function of energy density for various values of therelaxation time

εΓ

Γ

Fig.3 illustrate that in case EoS obtained by solving (10) corresponds toMaxwell construction.

0→Γ

We can see also that the evolution of system delays with increasing of proper time.

The softest points

Application to 1+1 D hydrodynamic model in the Cartesian coordinates

In case of 1+1 hydrodynamic model system (4) is reduced to

; 0))((

,0))((2

2

=∂+∂++∂+∂

=∂+∂++∂+∂

vvvPv

vvvPPPv

xtxt

xtxt

εγεε

εγwhere .

11 ,

2vuuv t

x

−== γ

Initial conditions:

[ ]

.2

)tanh(1)sgn(),(

,4

) tanh(1 ),(

fm/c, 0

210

2 210

0

0

kxkxtxv

kxktx

t

−+=

−−=

=

εε

GeV 40 toaccords GeV/fm 2 30 AElab ==ε

and are coefficients which are taken in agreement with initial radius of systemat Gev. 40 AElab =1k 2k

Space distributions of energy density at fixed times t .Fig. 5,6,7 correspond to phase transitions with various values of the relaxation time . Fig. 8 illustrates the expansion of fireball without phase transition.

Γ

fm/c001.0=Γ fm/c5.0=Γ

fm/c1=Γ

Fig. 6

Fig. 7

Fig. 5

Fig. 8

Phase transition slows down evolution of the system.

pure hadronphase

Fig. 10: dependence of 1D volume-averaged energy density on time for various values of the relaxation time Γ

Fig. 9: time-dependence of one dimensional volume of whole system for various relaxation times Γ

The total volume of expanding system weakly depends onrelaxation time (Fig. 9)

Fig.11: dependence of 1D volume-averagedpressure on time for various relaxation times Γ

Fig. 12: 1D volume-averaged pressure asa function of 1D volume-averaged energydensity for various values of the relaxation time Γ

When relaxation time goes to 0, effective EoS is similar to Maxwell construction (Fig. 12).

Γ

Fig. 13: little 1D volume-averaged energy density vs. time for various relaxationtimes Γ

Fig. 14: little 1D volume-averaged pressure vs. time for various relaxation times Γ

Fig. 15: little 1D volume-averaged pressure asa function of little 1D volume-averaged energydensity for various values of the relaxation time Г .

Obtained as implicit function by usingtime-dependencies (Fig.13) and (Fig.14).

Γ

)(tε)( tP

It reproduces Maxwell constructionfor Г=0.001 fm/c more precisely thaneffective EoS from Fig.12

Fig.16: Freeze out profiles for various values of the relaxation time .

Freeze out energy density

Γ3

fr GeV/fm5.0=ε

SummaryDynamical model of the first order phase transitionin the relaxation time approximation has been constructed. Existence of phase transition delays evolution of system. The experimental observables which may besensitive to the dynamics of phase transition:Dileptons and direct photons (dileptons from mixed

phase, see V.D. Toneev’s lecture).

Finite baryon chemical potential,3D calculations with a realistic EoS are tobe done.