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.
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.
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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.