Pis'ma v ZhETF, vol.90, iss.4, pp. 245-249
© 2009 August 25
Hydrodynamical description of a hadron-quark first-order phase
transition
V. V. Sioiov+*1), D.N. Voskresensky+v + GSI, D-64291 Darmstadt, Germany *Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia v Moscow Engineering Physics Institute, 115409 Moscow, Russia Submitted 16 June 2009
Solutions of hydrodynamical equations are presented for the equation of state of the Van der Waals type allowing for a first-order phase transition. As an example we consider the hadron-quark phase transition in heavy-ion collisions. It is shown that fluctuations dissolve and grow as if the fluid is effectively very viscous. In the vicinity of the critical point even in spinodal region seeds are growing slowly due to viscosity, surface tension and critical slowing down. These non-equilibrium effects prevent enhancement of fluctuations in the near-critical region, which in thermodynamical approach is frequently considered as a signal of the critical endpoint in heavy-ion collisions.
PACS: 25.75.Nq, 64.10.+h, 64.60.Bd
There are many phenomena, where first-order phase transitions occur between phases with different densities. Description of such phenomena should be similar to that for the gas-liquid phase transition. Thereby it is worthwhile to find corresponding solutions of hydrodynamical equations. Though some simplified analytical [1, 2] and fragmentary two-dimensional numerical solutions [3] have been found, many problems remain unsolved. In nuclear physics different first-order phase transitions (e.g., to pion, kaon condensates and to the quark state) may occur in neutron stars [2, 4] and in heavy-ion collisions [5, 6]. At low energies gas-liquid transition occurs [5]. It is also expected that at finite baryon density the hadron - quark gluon plasma (QGP) phase transition, which might manifest itself in violent nucleus-nucleus collisions, is of the first-order [6]. The hydrodynamical approach is efficient for description of heavy-ion collisions in a broad energy range (e.g. see [7, 8, 6]).
In this letter the dynamics of a first-order phase transition is described by equations of non-ideal non-relativistic hydrodynamics: the Navier-Stokes equation, the continuity equation, and general equation for the heat transport. We solve these equations numerically in two spatial dimensions, d = 2, and analytically for arbitrary d in the vicinity of the critical point. Then we perform estimations for the case of the hadron - QGP transition.
The best known example to illustrate principal features of a first-order phase transition is the Van der Waals fluid. The pressure is given by Pyw[n,T] = = nT¡(1 — bn) — n2a, where T is the temperature, n is the density of a conserving charge (e.g., the baryon charge), parameter a governs the strength of a mean field attraction and b controls a short-range repulsion.
We expand the quantities entering EoS and equations of hydrodynamics near a reference point (pT,TT) chosen somewhere in the vicinity of the critical point on the plane P(p,T), where p = mn is the mass density, to is the mass of the constituent. Assuming smallness of the velocity u(r, r) of the seed we linearize hydrodynamical equations in u, density 5p = p — pr and temperature ST = T —TT. Applying then operator "div" to the Navier-Stokes equation and taking z = divu from the continuity equation we obtain [1, 2]:
825p dt2
= A
SP + P71 (ànr + Cr)^
(1)
d = 2(d-l)/d, SP = P — P[pr,Tr] = is the pressure expressed through the free energy F for slightly inhomogeneous configurations; and £r are shear and bulk viscosities; A = d21 + ... + d2d.
Note that derived Eq. (1 ) differs from the phenom-enological Landau equation for the nonconserving order parameter dt<f> = ^7(SF/S</>), 7 = const, and from equations used for the description of the dynamics of firstorder phase transitions in heavy-ion collisions [9] and in relativistic astrophysical problems [10]. The difference with the Landau equation disappears, if one sets
^e-mail: V.Skokovôgsi.de
zero the square bracketed term in the r.h.s. of Eq. (1). From the first glance, such a procedure is legitimate, if space-time gradients are small. However for a seed, being prepared in a fluctuation at t = 0 with a distribution 5p(t = 0,r) = 5p(0,r), the condition d5p(t,r)/dt\t=o — 0 should also be fulfilled (otherwise there appears a kinetic energy term). Two initial conditions cannot be simultaneously fulfilled, if the equation contains time derivatives of the first-order only. Thus, there exists an initial stage of the dynamics of phase transitions (t < iinit)j which is not described by the standard Landau equation.
For low velocities the heat transport is described by the heat conductivity equation cv = kAT, where k is the heat conductivity and cv is the specific heat. Time scale of the temperature relaxation is tx = R2{tT)cv / k, where R(t) is the size of the seed. On the other hand, time scale of the density relaxation, following Eq. (1), is tp oc R (we show below that a seed of rather large size grows with constant velocity). Evolution of the seed is governed by the slowest mode. When sizes of seeds begin to exceed the value -Rfog, where Rfog is the size at which tx = tp, the growth is slowed down. Thus number of seeds with the size R ~ Rfog grows with time and there appears a metastable state called the fog.
We will consider phase transition for the system at fixed values of T and P at the boundary. For further convenience we choose pr = pcr, Tr = Tcr and expand the Landau free energy in 5p and ST:
SF = [ — J Pi
Then
c[V(5p)]2 A {5p)A Xv2(Sp)2
eSp
(2)
SF = FL[p,T]^FL[pI,TI]. q T i
v j. cr
a =
8 n.
b =
3 n.
%m2T HT
v2 = = 4\5T\n2crm2, T = (T - Tcr)/Tc,
A =
2 ab 3ab
9 Tr.
2to3 16 n2 to3 '
e = ncr(pi - /¿2),
where pi and /¿2 are chemical potentials of initial and final configurations (at fixed T and P at the boundary of the system). Maximum value emax = V^Tcrncr\ST\3^2■ In dimensionless variables Sp = v\j), = ar*//, i = 1, • • • , d, t = t/to, we arrive at
1= (2c/(Xv2))1/2, t0 = 2(~dVl
Q/(Xv2pI),
e= 2e/(Xv3), = 4/(3^3), /3 = cp2/[dVl + Cr]2-
Thus I oc \5T\-1/2 and to oc IJTI-1. With A' = A to2, v = v/m, r] = rj/m, ( = (/to, c = c/m, the dependence on the mass to can be excluded from all values in Eq. (3). Note that in (3) A£e = 0. We retained this term for convenience since then solutions (3) yield correct asymptotic for uniform configurations.
There exists an opinion, cf. Ref. [11], that, if at some incident energy the trajectory passes in the vicinity of the critical point, the system may linger longer in this region due to strong thermodynamical fluctuations resulting in the divergence of susceptibilities that may reflect on observables. Contrary, we argue that fluc-tuational effects in the vicinity of the critical point in heavy-ion collisions can hardly be pronounced, since all relevant processes are proved to be frozen for ST —t 0, while the system passes this region during a finite time.
To describe configurations of different symmetry we search two-phase solution of Eq. (3) in the form [1, 2],
t/> = =Ftanh[£ — £o(t)] + e/4,
(4)
£ = y/Q + £2 + for droplets/bubbles (dso 1 = 3), C = Vli+ci for rods №01 = 2) and £ = £1 = x/l for kinks (dsoi = 1) in d = 3 space. For c > 0 upper sign solution describes evolution of droplets (or rods and kinks of liquid phase) in a metastable super-cooled vapor medium. The lower sign solution circumscribes then bubbles (or kinks and rods of gas phase) in a stable liquid medium.
The boundary layer has the length |£ — £o(t)| ~ 1. Outside this layer corrections to homogeneous solutions are exponentially small. Considering motion of the boundary for £0 (r) > 1 we may put £ ~ £0 (r) in (4). Then keeping only linear terms in e in Eq. (3), we arrive at equation
/3d2 Co 2
dr2 =±2C'
4
'Sol
Co (r)
d£o dr
(5)
Substituting (4) in (2) we obtain
6F[£0] =
27T3/2A3-d-°lXvHd">1
r(dsol/2)F(l
(3 - dsol)/2)pr
2^,01-1/3
(6)
2A is the diameter, height of cylinder and the length of the squared plate for dsoi =3,2 and 1, respectively; F is the Euler F-function. The first term in (6) is the volume term and the second one is the surface contribution, SFsnif. At fixed volume in d = 3 space, the surface contribution for droplets/bubbles is smaller than for rods and slabs. Thereby if a seed prepared in a fluctuation
Hydrodynamical description of a hadron-quark first-order phase transition
247
is initially nonspherical it acquires spherical form with passage of time. Surface term is SFsurf = aS, S is the surface of the seed, a is the surface tension, and the gra-
dient term in (2) is then №sgurrafd = =
' surf
Thus we are able to find relations:
\SFn
2 surf-
ji _
GT^n^lSTl1^2
. There are two di-
mensionless parameters in (3) and (5): c and /3. The value c distinguishes metastable and stable state minima in the Landau free energy, ¡3 = (32Tcr)-1 [drjj + Qj^alm controls dynamics. For ¡3 -C 1 one deals with effectively viscous fluid and at ¡3 1, with perfect fluid. Thus the smaller surface tension, the effectively more viscous is the fluidity of seeds.
Using Eq. (5) we can consider analytically several typical solutions for evolution of seeds of stable phase in metastable matter.
1) Short time evolution of a seed. For small r (initial stage) using Taylor expansion in r and assuming zero initial velocity, — 0, we obtain
R(t) ~R0 + (wt2/2) [1 - 2i/(3i0/?)]
valid for t « t « iinit =
oc
fx" --
(di7r+Cr)|iT|
acceleration
. Initial stage of the process proceeds with
w = (dSoi - 1)A«2 (Ro - Rcr)/(RoRcr),
which changes sign at the initial size Ro = Rcr, where
Rcr = (dsol - l)v2V2cX/(3|c|), Rcr(emax) oc 1/\5T\1/2,
is the critical size. Seeds with Rq < Rcr shrink, while seeds with Rq > Rcr grow. For seeds with \Ro — Rcr\ -C RCr the size changes very slowly (w oc oc |5T|(-Ro — Rcr)/Rcr)- F°r undercritical seeds of a small size, w oc — \5T\/Ro- S
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