Viscosity and thermal conductivity effects at first-order phase transitions in heavy-ion collisions

# Viscosity and thermal conductivity effects at first-order phase transitions in heavy-ion collisions

## Abstract

Effects of viscosity and thermal conductivity on the dynamics of first-order phase transitions are studied. The nuclear gas-liquid and hadron-quark transitions in heavy-ion collisions are considered. We demonstrate that at non-zero thermal conductivity, , onset of spinodal instabilities occurs on an isothermal spinodal line, whereas for istabilities take place at lower temperatures, on an adiabatic spinodal.

## I Introduction

There are many phenomena, where first-order transitions occur between phases of different densities. In nuclear physics various first-order phase transitions may take place in the Early Universe, in heavy-ion collisions and in neutron stars (e.g., pion condensation, kaon condensation, deconfinement and chiral phase transitions), see reviews  (1); (2); (3). At low collision energies the nuclear gas-liquid (NGL) first-order phase transition is possible (1); (4); (5); (6). At high collision energies the hadron-quark gluon plasma (HQGP) first-order transition may occur (see e.g. Ref. (2)). Within a hydrodynamical approach dynamical aspects of the NGL and HQGP transitions were recently studied in Refs. (7); (8); (9); (10); (11). An important role of effects of non-ideal hydrodynamics was emphasized. In this talk, we review some results obtained in Refs. (7); (8); (10).

## Ii General setup

We are interested in a description of long wavelength phenomena at a first-order phase transition. Thereby we accept an assumption that the velocity of a fluctuation (seed) is much slower than the mean thermal velocity. Then a description is possible in the framework of the standard system of equations of non-relativistic non-ideal hydrodynamics: the Navier-Stokes equation, the continuity equation, and equation for the heat transport:

 mn[∂tui+(u∇)ui] = −∇iP+∇k[η(∇kui+∇iuk−2dδikdivu)+ζδik% divu], (1) ∂tn+div(nu) = 0, (2) T[∂s∂t+div(su)] = div(κ∇T)+η(∇kui+∇iuk−2dδikdivu)2+ζ(divu)2, (3)

is the density of the conserving charge (here the baryon density), is the (baryon) mass, is the pressure, and are the shear and bulk viscosities, is the dimensionality of space, is the temperature, is the entropy density, is the thermal conductivity.

The simplest (yet non-trivial) example illustrating principal features of a first-order phase transition in a mean-field approximation is the Van der Waals fluid. The adiabatic trajectories, const, for an expansion of a uniform fireball to vacuum, are shown in Fig. 1. The super-cooled vapor (SV) and the overheated liquid (OL) regions are between the Maxwell construction (MC) and the isothermal spinodal (ITS) curves, on the left and on the right respectively. The adiabatic spinodal (AS) curve bounds the AS region from above. For , where corresponds to the value of the specific entropy at the critical point and the line with passes through the point at , the system traverses the OL state (the region OL in Fig. 1), the ITS region (below the ITS line) and the AS region (below the AS line). For the system trajectory passes through the SV state (the region SV in Fig. 1) and the ITS region.

All thermodynamic quantities can be expanded near an arbitrary reference point , or . Considering the problem in variables it is convenient to take the reference point in the vicinity of the critical point but outside the fluctuation (critical) region (, ) assuming that the latter is very narrow. Note that even in the fluctuation region, the mean-field treatment can be used provided one considers the system at time scales, being shorter than the scale responsible for a development of long-scale critical fluctuations. Thus we further put .

Let us construct a generating functional in the variables , , also called the Landau free energy, such that :

 δFL=∫d3xncr{cm[∇(δn)]22+λm3(δn)44−λv2m(δn)22−ϵδn}, (4)

where is expressed through the pressure at the MC. The maximum of the quantity is . For the sake of convenience, we will use normalized to its maximal value, where . The first term in Eq. (4) is due to the surface tension, , is the surface of the seed. For the Van der Waals equation of state:

 v2(T)=−4δTn2crm2Tcr,λcr=9f016Tcrn2crm3,σ=σ0|δT|3/2T3/2cr,σ0=32mn2crTcrc. (5)

Using Eq. (2), we rewrite Eq. (1) in the dimensionless variables , , , as

 −β∂2ψ∂τ2=Δξ(Δξψ+2ψ(1−ψ2)+˜ϵ−∂ψ∂τ), (6) l=(2cλv2)1/2,t0=2(˜dηr+ζr)λv2ρr,˜ϵ=2ϵλv3,β=cρ2r(˜dηr+ζr)2,

. Only linear terms in the velocity were kept in deriving this equation. Since , processes in the vicinity of the critical point are proven to be very slow. This is known as the critical slowing down phenomenon.

Eq. (6) should be supplemented by Eq. (3) for the heat transport, which owing to Eq. (2) after linearization simplifies to

 Tr[∂tδs−sr(nr)−1∂tδn]=κrΔδT. (7)

The variation of the temperature is related to the variation of the entropy density by where is the specific heat density.

Note that Eq. (6) differs from the standard Ginzburg-Landau equation exploited in phenomenological approaches. The difference disappears, if one sets the bracketed term in the r.h.s. of Eq. (6) to zero. Superficially this simplification is legitimate, if space-time gradients are small. However, for a seed prepared in a fluctuation at with a distribution , the initial condition should be fulfilled. Otherwise, because of a positive kinetic energy contribution the probability of fluctuations is suppressed. On the other hand, the above initial conditions cannot be simultaneously fulfilled, if a differential equation is of the first-order with respect to time derivative. Therefore, there exists an initial stage of the dynamics of seeds (), which is not described by the standard Ginzburg-Landau equation.

The time scale for the relaxation of the density following Eq. (6) is , where is the size of a seed (as we will show below, at some stage overcritical seeds grow with constant velocity), and the time scale for the relaxation of the entropy/temperature, following (7), is

 tT=R2cV,r/κr∝R2. (8)

The evolution of a seed is governed by the slowest mode. Thus, for , i.e for ( is the typical seed size at which ), dynamics of seeds is controlled by Eq. (6) for the density mode. For seeds with sizes , exceeds and growth of seeds is slowed down. Thereby, the number of seeds with the size may increase with time. Estimates (7); (8) show that for the HQGP phase transition fm and for the NGL transition fm , where is the fireball size at the freeze out, is the fireball evolution time till freeze out. Thus, thermal conductivity effects may manifest themselves in heavy-ion collision dynamics.

Note that seeds of a new phase are produced in an old phase owing to short-scale fluctuations. Fluctuations are not incorporated in the above hydrodynamical equations defined in terms of mean-field variables. Contributions of short-scale fluctuations can be simulated by a random force induced in Eqs. (6), (7), cf. (12).

There are only two dimensionless parameters in Eq. (6), and . The parameter is responsible for a difference between the Landau free energies of the metastable and stable states. For (isothermal stage), and dependence on the latter quantity disappears because of . Therefore, dynamics is controlled only by the parameter , which characterizes inertia (enters together with the second derivative in time). This parameter is expressed in terms of the surface tension and the viscosity as

 β=(32Tcr)−1[˜dηr+ζr]−2σ20m. (9)

The larger viscosity and the smaller surface tension, the effectively more viscous (inertial) is the fluidity of seeds. For one deals with the regime of effectively viscous fluid and at , with the regime of perfect fluid. For the NGL phase transition we estimate . For the HQGP phase transition , even for the very low value of the ratio  (7); (8). Thus one deals with effectively very viscous fluidity of density fluctuations in both NGL and HQGP transitions.

## Iii Dynamics of seeds in metastable area

Let us consider the stage . For an expanding system we have to assume that typical time for the formation and evolution of a fluctuation of our interest is much smaller than the typical fireball expansion time . Let us consider the situation, when at very slow expansion with const spatially quasi-uniform spherical fireball of a large radius enters either the OL- state or the SV- state (see the corresponding curves in Fig. 1). In case , i.e. in the vicinity of the critical point , solution (6) describing dynamics of the density in the fluctuation is presented (in the dimensional form) as (7); (8):

 δn(t,r)≃v(T)m[±thr−Rn(t)l+ϵ2λcrv3(T)]+(δn)cor, (10)

where the upper sign corresponds to the evolution of bubbles and the lower one to the evolution of droplets, , and the solution is valid for . The correction is responsible for the exact baryon number conservation. Considering in the vicinity of a bubble/droplet boundary we get equation describing evolution of the seed size:

 βt202d2Rndt2=3ϵ2λcrv3(T)−2lRn−t0ldRndt. (11)

Following this equation a bubble of an overcritical size of the stable gas phase, or respectively a droplet of the liquid phase, been initially prepared in a fluctuation, will grow. On the early stage of the evolution the size of the bubble/droplet grows with an acceleration. Then it reaches a steady grow regime with a constant velocity . In the interior of the seed . The correction is very small for .

Substituting Eq. (10) to Eq. (7) for const we obtain

 δs=(∂s∂n)T{v(T)m[±thr−Rn(t)l+ϵ2λcrv3(T)]+(δn)cor}. (12)

While the temperature is constant in the interior and exterior of the seed, the entropy and the density are different in the interior and exterior regions. Nevertheless the approximation of a quasi-adiabatic expansion of the system might be used even, when the system reaches metastable region, provided the gas of seeds is rare, or is small.

In Fig. 2 we demonstrate numerical solutions of the hydrodynamical equations in two dimensions () at the stage . We take and compute the configuration for MeV and for (effectively large viscosity). The choice  MeV, ( fm) is relevant for the HQGP phase transition. We see that undercritical seeds (disks) dissolve rather rapidly (typical time is several fm) but overcritical seeds grow very slowly. Similar solutions exist for bubbles. Therefore, one can hardly expect to observe a manifestation of large size droplet/bubble remnants in heavy-ion collisions.

The limit is specific. In case , i.e. in the vicinity of the point , corresponding solutions of Eq. (6) describing dynamics of the density can be presented in the form (10) with the only difference that should be replaced by , by and by . Dynamics of is determined by Eq. (11), where one should replace values calculated at to the corresponding values at . From Eq. (6) we obtain with given by Eq. (10). From known values and we can define . Therefore, not only the density, , but also the entropy density, , and the temperature, , vary in the surface layer and exhibit different values inside and outside a seed. Contrary, the value remains constant.

## Iv Instabilities in spinodal region

In this section the “r”-reference point can be taken arbitrary, therefore, we suppress the subscript “r”. To find solutions of the hydrodynamical equations we put, cf. (7); (8),

 δn=δn0exp[γt+ipr],δs=δs0exp[γt+ipr],T=T>+δT0exp[γt+ipr], (13)

where is the temperature of the uniform matter. From the linearized equations of non-ideal hydrodynamics we find the increment, ,

 γ2=−p2[u2T+(~dη+ζ)γmn+cp2+u2~s−u2T1+κp2/(cVγ)], (14)

and are speeds of sound at constant temperature and entropy, respectively. Eq. (14) has three solutions. Expanding the solutions for small momenta (long-wave limit) we find

 γ1,2 = ±iu~sp+[κcV(u2Tu2~s−1)−~dη+ζmn]p22, (15) γ3 = −κu2Tp2u2~scV[1−u2T−u2~su2~su2T(c+κu4Tu2~sc2V−(~dη+ζ)κu2TmncVu2~s)p2]. (16)

The solutions correspond to the sound mode in the long wavelength limit, whereas describes the thermal transport mode. Below the ITS line (and above the AS line) , , solutions correspond to an oscillation and damping, whereas describes an unstable growing mode. Below the AS line, since there and , the modes exchange their roles: the sound modes become unstable, while the thermal mode is damped.

For sufficiently high thermal conductivity and not as small , , within the ITS region the most rapidly growing mode (, and ) is the density mode:

 γm=γ(1,2)m=(−u2T)mncr(2√β+1)(~dη+ζ),p2m=(−u2T)√β(2√β+1)c.

Using these expressions we may rewrite the condition of a high thermal conductivity as Typical radius of structures is , it decreases with increasing .

The amplitudes of the temperature and density are related by

 δT0=δn0Ts[1−n(∂s/∂n)T/s]cVn[1+κ/(√ccV)]. (17)

Therefore, the assumption of spatial homogeneity of the system fails right after the ITS region is reached. An aerosol (mist) of bubbles and droplets is formed for a typical time . For the Van der Waals equation of state we find that , i.e. the temperature is larger in denser regions. However, the amplitude of the temperature modulation is rather small for

Let us consider the case of zero shear and bulk viscosities and non-zero, but small thermal conductivity. For , i.e. slightly below the ITS line, we get

 γm=γ(3)m≃κu4T4ccVu2~s,p2m≃−u2T/(2c). (18)

Therefore, in both considered cases the instability occurs at the ITS line. However, for the most rapidly growing mode corresponds to and in the opposite limit , to .

In Fig. 3 we show the time evolution of the density wave amplitudes given by the first Eq. (13), for an undercritical value of , (left panel), and for an overcritical value (right panel) for the same choice of the parameters ,  MeV, , as in Fig. 2. We see that the evolution is more rapid compared to the configuration presented in Fig. 2 and the characteristic time scale  fm is comparable with the fireball expansion time . Thus we may conclude that in heavy-ion collisions during expansion of the fireball the system may linger in QGP phase at . This means that the equilibrium value of the critical temperature of the phase transition might be significantly higher than the value which may be manifested in growth of fluctuations in experiments with heavy ions. Fluctuations grow more rapidly with decrease of below ,  fm. Far from the critical point rapid grows of fluctuations reminds effect of a warm champagne. There are prospects for observing specific signatures of fluctuations with a typical size, defined by , in heavy-ion collisions. Fluctuations of this kind might be distinguishable from ordinary statistical fluctuations.

The limit is again specific. Below the ITS line and above the AS line, the thermal mode , which drives the system towards equilibrium for a small thermal conductivity, does not exist for . Therefore, the evolution in the spinodal region is entirely governed by adiabatic sound excitations. The increment is given by Eq. (14) with replaced by :

 γ2=−p2[u2~s+(~dη+ζ)γmn+cp2]. (19)

Therefore, contrary to the case , instability appears, when the system trajectory crosses the AS line rather than the ITS line. The value for the particular case of the Van der Waals fluid in the vicinity of the point (taken as the reference point). This result holds also in the case of ideal hydrodynamics, where in addition to , the viscosity coefficients (, ) are zero. From Eq. (7), we find

 δs0=δn0s/n,δT0=δn0Ts[1−n(∂s/∂n)T/s]cVn. (20)

Thereby, the temperature is modulated similar to the entropy density. For the Van der Waals equation of state we find that . The amplitude of the temperature modulation is larger than in case of , see Eq. (17).

Concluding, for any the solutions of Eq. (14) result in the onset of the instability already for (i.e. below the ITS line rather than below the AS line). Since in reality is indeed nonzero, spinodal instabilities start to develop when the trajectory crosses the ITS line rather than the AS, i.e. at significantly higher temperatures. This favors an observation of signals of the spinodal decomposition in the HQGP phase transition in heavy-ion collisions.

We expect that owing to a manifestation of non-trivial fluctuation effects (especially, of the spinodal decomposition at first-order hadron-quark transition) a non-monotonous behavior of different observables as function of collisional energy may be observed. This experimental analysis will be possible at RHIC, FAIR and NICA. Owing to properties of spinodal decomposition a manifestation of specific structures with typical spatial size may serve as a promising signal of the QCD first-order phase transition.

Acknowledgments. This work was supported in part by the DFG grant WA 431/8-1.

FIGURE CAPTIONS

1. The phase diagram of the Van der Waals equation of state, -plane. The bold solid, dashed and dash-dotted curves demonstrate the boundaries of the Maxwell construction, the spinodal region at const and const, respectively. The short dashed lines show adiabatic trajectories of the system evolution: the curve labeled passes through the critical point; , through the maximum pressure point on the plane.

2. The isotherm for the pressure as a function of the density, with initial and final configurations shown by the dots (left column). The dashed vertical line shows the MC. The initial state represents the stable liquid phase disk () in metastable SV. Middle column demonstrates the time evolution of the density of the overcritical liquid disk. The numbers near the curves (in ) denote time moments; , , fm. Right column, the same for the undercritical liquid disk.

3. Time evolution of the wave amplitudes defined as normalized to the amplitude of the initial disturbance. Solid line is for effectively small viscosity () and dash line, for the large viscosity (). Left panel: the undercritical wave number (growing modes). Right panel: the overcritical value (oscillation modes for large and damped modes for small ). Other parameters are taken to be the same, as in Fig. 2.

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