Hydrodynamical description of a hadron-quark first-order phase transition.
Solutions of hydrodynamical equations are presented for the equation of state of the Var der Waals type allowing for the first order phase transition. Attention is focused on description of 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. Even in spinodal region germs are growing slowly due to viscosity and critical slowing down. This prevents enhancement of fluctuations in the near-critical region, which is frequently considered as a signal of the critical point in heavy ion collisions.
pacs:25.75.Nq, 64.60.Bd, 64.10.+h
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 PS (); MSTV90 () and fragmentary two-dimensional numerical solutions Onuki () 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 MSTV90 (); Glendenning () and in heavy ion collisions Randrup (); Shuryak:2008eq (). At low energies gas-liquid transition occurs Randrup (). 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 Shuryak:2008eq (). The hydrodynamical approach is efficient for description of heavy-ion collisions in a broad energy range (e.g. see SG (); Romatschke:2007mq (); Shuryak:2008eq ()).
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, , and analytically for arbitrary 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 where is the temperature, is the density of a conserving charge (e.g., the baryon charge), parameter governs the strength of a mean field attraction and controls a short-range repulsion. In practice we use a modified Van der Waals (mVW) equation of state (EoS): where the function is chosen so that at the critical density and the critical temperature . We use , and the pre-factor is chosen to reproduce ideal gas EoS for sufficiently low and high . This modification allows us to parameterize the EoS with two minima in the free energy, being convenient for analytical treatment of the problem near the critical point . We expand the quantities entering EoS and equations of hydrodynamics near a reference point () chosen somewhere in the vicinity of the critical point on the plane , where is the mass density, is the mass of the constituent. Assuming smallness of the velocity of the germ we linearize hydrodynamical equations in , density and temperature . Applying then operator ”div” to the Navier-Stokes equation and taking from the continuity equation we obtain PS (); MSTV90 ():
. The pressure is expressed through the Helmholtz free energy for slightly inhomogeneous configurations; and are the first (shear) and second (bulk) viscosities;
Note that thus derived Eq. (1) differs from the phenomenological Landau equation for the nonconserving order parameter , and from equations used for the description of the dynamics of first-order phase transitions in heavy ion collisions Gavin () and in relativistic astrophysical problems astro (). The difference with the Landau equation disappears, if one sets 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 germ, being prepared in a fluctuation at with a distribution , the condition 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 (, which is not described by the standard Landau equation.
For low velocities the heat transport is described by the heat conductivity equation where is the heat conductivity and is the specific heat. Time scale of the temperature relaxation is where is the size of the germ. On the other hand, time scale of the density relaxation, following Eq. (1), is (we show below that a germ of rather large size grows with constant velocity). Evolution of the germ is governed by the slowest mode. When sizes of germs begin to exceed the value , where is the size at which , the growth is slown down. Thus number of germs with the size grows with time and there appears a metastable state called the fog.
For further convenience we choose , and expand the Helmholtz free energy in and :
. Then Introducing dimensionless variables , , , , we arrive at equation
Thus and .
There exists an opinion, cf. Ref. Stephanov (), 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 divergence of susceptibilities that may reflect on observables. Contrary, we argue that fluctuational 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 , while the system passes this region during a finite time.
for droplets/bubbles (), for rods () and for kinks () in space. For 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 . Outside this layer corrections to homogeneous solutions are exponentially small. Considering motion of the boundary for we may put in (4). Then keeping only linear terms in in Eq. (3), we arrive at equation for :
is the diameter, height of cylinder and the length of the squared plate for and , respectively; is the Euler -function. The first term in (6) is the volume term and the second one is the surface contribution, . At fixed volume in space, the surface contribution for droplets/bubbles is smaller than for rods and slabs. Thereby if a germ prepared in a fluctuation is initially nonspherical it acquires spherical form with passage of time. Surface term is , is the surface of the germ, is the surface tension, and the gradient term in (2) is then Thus we are able to find relations: There are two dimensionless parameters in (3) and (5): and . The value distinguishes metastable and stable state minima in the free energy, controls dynamics. The larger viscosity and the smaller surface tension, the effectively more viscous is the fluidity of germs. For one deals with effectively viscous fluid and at , with perfect fluid.
At hand of Eq. (5) consider analytically several typical solutions for germ evolution. Consider evolution of germs of stable phase in metastable matter.
1) Short time evolution of a germ. For small (initial stage) using Taylor expansion in and assuming zero initial velocity, , we obtain
valid for and . Initial stage of the process proceeds with acceleration
which changes sign at the initial size where
is the critical size. Germs with shrink, while germs with grow. For germs with the size changes very slowly (). For undercritical germs of a small size, . Slabs of stable phase, being placed in a metastable medium, grow independently of what was their initial size. Note that the same value follows from minimization of the free energy (6).
2) Long time evolution of a large germ. For , we may drop the term in the l.h.s of Eq. (5). For , surface effects become unimportant and we arrive at the solution
Germs grow with constant velocity. The time scale for the growth of the germ with size is Asymptotic regime is reached at very large values of time, provided the system is near the critical point.
3) Long time evolution of a small germ. Describing germs of a small size (, ) for , we can drop the term in (5). Then solution acquires the form
The time scale at which the initial germ of a small size dissolves is, and is . Thus, fluctuations of sufficiently small sizes are easily produced and dissolve rapidly.
4) Fluctuations in spinodal region. Let the system be driven to a spinodal region where fluctuations of even infinitesimally small amplitudes and sizes may grow into a new phase. To demonstrate this we take the free energy to be close to its maximum (). Then we linearize Eq. (3) dropping term. Setting is an arbitrary but small real constant, we find two solutions,
Growing modes correspond to the choice of ””-sign and . The time scale at which an aerosol of germs develops is corresponds to . For an effectively large viscosity () there are two solutions: the damped one, and the growing one for . The most rapidly growing mode is , . The time scale characterizing growth of this mode is The typical size of germs, increases with an increase of the viscosity. For both modes are damped. In the case of an effectively small viscosity () we get and The time scale characterizing growing modes, does not depend on the viscosity in this limit. The size scale of germs is Modes with oscillate and do not grow into a stable phase.
For the description of the hadron–QGP first-order phase transition we take values MeV, , as they follow from lattice calculations, see Aoki:2005vt (). Parameters of the EoS are then as follows: , fm, , , , where is the effective quark mass, . Further we obtain fm (radius of confinement) for MeV. If one used MeV, one would estimate fm.
Next we use at near , , as it follows from the lattice data Aoki:2005vt (). Assuming the minimal value of the viscosity MeV/fm, we evaluate for MeV, that corresponds to the limit of effectively very large viscosity. Even for MeV, MeV we would get . Note that following Kharzeev () the bulk viscosity diverges in the critical point. If were so (), the quark-hadron system would behave as absolutely viscous fluid, like glass, in near critical region. Contrary, Ref. SR () argues for a smooth behavior of the bulk viscosity.
With we further estimate fm, and . Typical time for the formation of the aerosol is fm, and typical size of germs in aerosol is fm. Only fm proves to be small (excluding quite small ). Critical slowing down that limits growing of the meson correlation length was discussed in Berdnikov ().
For the thermal conductivity we use an estimation taking to recover the relation between values of and for nuclear gas-liquid phase transition at low energies GIK (). The scale of the heat transport time is fm. Using that fm, we obtain . The value proved to be very small ( fm). However, time scale is rather long. Therefore, the system most probably would have no time to fully develop a fog-like state in a hadron-quark phase transition in heavy ion collisions.
For the system in the vicinity of the critical point all estimated time scales (except ) are very large. If the system trajectory paths rather far from the critical point (), all time scales, except , become less than the typical life-time of the fireball ( fm at RHIC conditions). Reynolds numbers are , being much smaller than the critical value (). Thereby, turbulence regime is not reached.
We solved numerically the general system of equations of nonideal hydrodynamics for . To illustrate the results we consider dynamics of overcritical and undercritical germs (disks) in infinite matter taking initial density profile as , and are densities in stable and metastable homogeneous phases, respectively.
In Fig. 1 we show the time evolution of a liquid disk (upper panel) and a gas disk (lower panel) for . In the middle column we show dynamics of an initially overcritical germ with and in the right column, of undercritical germ , fm. The time snapshots are shown in Figure in units L. The configuration is computed for values of kinetic parameters MeVfm and . We see that in case (middle column) disks slowly grow with the time passage. For overcritical discs the initially selected distribution acquires the tanh-like shape, see (4), only for . Initial disks of a small size practically disappear for fm. Due to the matter admission to the disk surface and the shape reconstruction, the density decreases in the liquid disk neighborhood below the value of the density in the homogeneous metastable matter and it increases in the gas disk surrounding above the value of the density in the homogeneous metastable matter (see the middle column).
In Fig. 2 we demonstrate time evolution of the wave amplitudes, , for an undercritical value of the wave number (left panel) and for an overcritical value (right panel). In case of the overcritical and effectively small viscosity () we demonstrate change of the amplitude in the -periods of the oscillation. Such a behavior fully agrees with that follows from our analytical treatment of the problem.
Concluding, even in the spinodal region germs are growing slowly, if the system is somewhere in the vicinity of the critical point. Thus in heavy ion collisions the expanding fireball may linger in the QGP state, until decreases below the corresponding equilibrium value of the temperature of the phase transition. There exists a belief that strongly coupled QGP state, represents almost perfect fluid Romatschke:2007mq (). We demonstrate the essential role of viscosity and surface tension in dynamics of first-order phase transitions, including the hadron-QGP one. Fluctuations in QGP (at a finite baryon density) grow and dissolve as if the fluid were very viscous. Variation of parameters in broad limits does not change conclusions.
We are grateful to B. Friman, Y.B. Ivanov, E.E. Kolomeitsev, J. Randrup, and V.D. Toneev for numerous discussions. This work was supported in part by the DFG project 436 RUS 113/558/0-3, and RFBR grants 06-02-04001 and 08-02-01003-a.
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