Dynamical growth of the hadron bubbles during the quark-hadron phase transition

PHYSICAL REVIEW D, VOLUME 63, 014012

Dynamical growth of the hadron bubbles during the quark-hadron phase transition

P. Shukla, A. K. Mohanty, and S. K. GuptaNuclear Physics Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India

~Received 6 March 2000; revised manuscript received 2 August 2000; published 7 December 2000!

The rate of dynamical growth of the hadron bubbles in a supercooled baryon free quark-gluon plasma isevaluated by solving the equations of relativistic fluid dynamics in all regions. For a nonviscous plasma, thisdynamical growth rate is found to depend only on the range of correlationj of order parameter fluctuation, andthe radiusR of the critical hadron bubble, the two length scales relevant for the description of the criticalphenomena. Further, it is shown that the dynamical prefactor acquires an additive component when themedium becomes viscous. Interestingly, under certain reasonable assumptions for the velocity of the sound inthe medium around the saddle configuration, the viscous and the nonviscous parts of the prefactor are found tobe similiar to the results obtained by Csernai and Kapusta and Ruggeri and Friedman~for the case of zeroviscosity!, respectively.

DOI: 10.1103/PhysRevD.63.014012 PACS number~s!: 12.38.Mh, 64.60.Qb

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I. INTRODUCTION

The phenomena of phase transition have attracted mresearchers from diverse areas due to many interestingcommon features that occur near the transition point.cently, a considerable amount of attention has been paithe study of relativistic heavy ion collisions where a phatransition is expected from the normal nuclear matter tdeconfined state of quarks and gluons@1#. The quark gluonplasma~QGP!, if formed, would expand hydrodynamicalland would cool down until it reaches a critical temperatuTc where a phase transition to hadron phase begins.though the order of such a phase transition remains ansettled issue, a considerable amount of work has been caout to understand the dynamics assuming it to be of fiorder and also assuming that the homogeneous nucleatiapplicable@2–4#. In the ideal Maxwell construction, the temperature of the plasma remains fixed atTc during the phasetransition until the hadronization gets completed. Howevif the hadronization proceeds through nucleation, it will nbegin atT5Tc due to the large nucleation barrier. The nucation of the hadron bubbles can begin only from a supcooled metastable state. If the amount of supercoois small, the nucleation rate@5# is computed from I5A exp(2DF/T) which gives the probability per unit timeper unit volume to nucleate a region of the stable phase~thehadron phase! within the metastable phase~the QGP phase!.HereDF is the minimum energy needed to create a critibubble and the prefactorA is the product of statistical andynamical factors. The statistical factorV0 is a measure ofboth the available phase space as the system goes ovesaddle and of the statistical fluctuations at the saddle relato the equilibrium states. The dynamical prefactork givesthe exponential growth rate of the bubble or droplet sitton the saddle.

In an earlier work, Langer and Turski@6# derived thedynamical growth rate (k) of the liquid droplet based onnonrelativistic formalism. Subsequently,k was derived bothby Langer and Turski@7# and Kawasaki@8# for a liquid-gasphase transition near the critical point, to be

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which involves the thermal conductivityl, the surface freeenergys, the latent heat per moleculel, and the density ofthe molecules in the liquid phasenl . The interesting physicsin this expression is the thermal conductivity which appeas an essential ingredient for the transportation of the laheat away from the surface region so that the dropletgrow. For a relativistic system like a baryon free quark gluplasma which has no net conserved charge, the thermalductivity vanishes. Hence, the above formula obvioucannot be applied to such systems. Therefore, CsernaiKapusta rederivedk for a baryon free plasma using earlieformalism of Langer and Turski@6#, but extending theirwork to the relativistic domain@9#. In the work of Langerand Turski@6#, k was derived by solving a set of linearizehydrodynamic equations in the liquid, vapor, and interfacregions. However, in the relativistic formalism, Csernai aKapusta mostly concentrated in the interfacial region. Thprimary motivation was to know the velocity profile in thsurface region which was then used to estimate the enflow across the surface. Then they used the condition thaenergy flux, which is to be transported outwards, shouldbalanced by the viscous heat dissipation as follows:

DvdR

dt52~4h/31z!vdv/dr, ~2!

whereR is the radius of the hadron bubble andv(r ) is theflow velocity just outside the surface of the bubble. Accoringly, they obtained an expression fork given by

k5

4sS 4

3h1z D

~Dv!2R3, ~3!

whereh andz are the shear and bulk viscosity coefficienrespectively andDv is the difference in the enthalpy densties of the plasma and the hadronic phases,v5e1p. The

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P. SHUKLA, A. K. MOHANTY, AND S. K. GUPTA PHYSICAL REVIEW D63 014012

above expression implies that energy flowvv is provided bythe viscous effects. There will be no bubble growth in tcase of an ideal plasma with zero viscosity. Thus, the viscity plays the same role as thermal conductivity in the casea relativistic fluid like the quark gluon plasma with zebaryon density. This approach has been extendedVenugopalan and Vischer@10# for the case of baryon richQGP where both viscous damping and thermal dissipaare significant. On the contrary, Ruggeri and Friedman~RF!@11# argued that the energy flow does not vanish in thesence of any heat conduction or viscous damping. Sincechange of the energy densitye in time is given in the lowvelocity limit by the conservation equation@9#, ]e/]t52¹•(vv) which implies that the energy flow}vv is alwayspresent. Therefore, following a different approach, Ruggand Friedman@11# derived an expression fork which doesnot vanish in the absence of viscosity, given by

k5S 2s

R3

vq

~Dv!2D 1/2

. ~4!

The viscous effects cause only small perturbations toabove equation. This result is in contradiction with the epression given in Eq.~3! according to which the hadrobubble will not grow in the limit of vanishing viscosity. Thdifference between the Csernai-Kapusta~CK! and Ruggeri-Friedman~RF! results are due to the technical differencesthe treatment of the pressure gradients and it needs fuinvestigation. Motivated by this, we rederivek using theCsernai-Kapusta formalism which is a relativistic generalition of the Langer-Turski~LT! procedure@12#. However, un-like Csernai-Kapusta, we solve the linearized hydrodynaequations in all regions namely the exterior quark region,interior hadron region as well as the interfacial or the surfregion. We found that in the limit of zero viscosity, ouprefactork depends only on two scale parameters, the crelation lengthj, and the critical radius of the hadron bubbR. We have also obtained the prefactor for a viscous medwhere it can be written in a simple way as the sum oviscous and a nonviscous terms. Interestingly, using cerassumption for the velocity of sound in the medium arouthe saddle configuration, the viscous and nonviscous comnents are found to be similiar to the results as obtainedCsernai and Kapusta@Eq. ~3!# and Ruggeri and Friedma@for zero viscosity, Eq.~4!# respectively.

The paper is organized as follows. We begin with a brreview of the Csernai-Kapusta and Turski-Langer formalidescribing the energy-momentum conserving equationsmotion in Sec. II. In Sec. III, we solve these equationsderive the dynamical prefactor. Finally, the conclusionspresented in Sec. IV.

II. THE RELATIVISTIC HYDRODYNAMICSFOR BARYON FREE PLASMA

In the case of relativistic hydrodynamics, we considerenergy densitye(r ,t) and the flow velocityv(r ,t) of thefluid as two independent variables that describe the dynam

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of the system. The equations of motion can be obtained frthe local conservation laws:

]mTmn5]mnm50. ~5!

HereTmn is the energy momentum tensor andnm representsthe baryon four vector. In the presence of viscosity,energy-momentum tensorTmn and baryon four vector currennm can be decomposed into an ideal and a viscous@13,14#:

Tmn5@~e1p!umun2pgmn#1tmn, ~6!

nm5num1nm. ~7!

Heree, p, andn are the energy density, pressure, and partnumber density. The fluid four velocity is given byum

5g(1,v) andtmn andnm are the dissipative corrections. Thform of the dissipative termstmn and nm depend on thedefinition of what constitutes the local rest frame of the fluThe four velocityum should be defined in such a way thata proper frame of any given fluid element, the energy andnumber densities are expressible in terms of other thermonamic quantities by the same formulas, when dissipative pcesses are not present. It is also necessary to specify whum is the velocity of energy transport or particle transpoAccordingly, there exist two definitions for the rest framone due to Landau and the other due to Eckart. In the Lanapproach,um is taken as the velocity of energy transportthat energy three fluxT0i vanishes in a comoving fram@13,14#. In the Eckart definition,um is taken as the velocityof the particle transport and the particle three current, ratthan the energy three flux vanishing in the fluid rest fra@14#. So in the Eckart definition of rest frame, the particfour vector can be written asnm5(n,0), whereas in theLandau definition of rest framenm5(n,n). Therefore, thetwo frames are related by a Lorentz transformation withboost velocityn/n. It is found that, due to ill-defined boosvelocity @15#, the energy three flux in the Eckart fram~which involves heat conductivityl) is not well defined asldiverges in the limit of chemical potentialm→0. On theother hand, in the Landau definition, heat conduction enas a correction to baryon flux. It was shown that, in spitethe divergence ofl, the correction to the baryon fluxnm isfinite @15#. Therefore, we will use the Landau definition fothe subsequent study and also we will assume a baryonplasma for simplicity. We can now write the equationsmotion from the conservation law]mTmn50 using theLandau definition@9,15#:

] te52¹•~vv!1O~v2!, ~8!

] t~vv!52¹•~vv^ v!2¹p

1¹S S 4

3h1z D¹•vD1O~v2!. ~9!

Herev5(e1p) andh andz are the shear and bulk viscosity coefficients respectively. We have also assumed thespeed limit whereg'1. Although, the fluid velocity is

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DYNAMICAL GROWTH OF THE HADRON BUBBLES . . . PHYSICAL REVIEW D 63 014012

small, the velocity of individual particles is large. Thus, texpressions for the energy density, pressure, etc., are takbe the same as that in the relativistic case.

Following Ref.@9#, Eq. ~9! can also be written in terms othe Helmholtz free energyf (e) and the usual gradient energ12 K(¹e)2 as

] t~vv!52¹•~vv^ v!2¹p81¹„~ 43 h1z!¹•v…1O~v2!,

~10!

where

2¹p852K~¹2e!¹e1] f

]e¹e. ~11!

The constantK is related to the surface tensions as

s5KE2`

`

drS de

dr D2

. ~12!

It can be noted by comparing Eq.~10! with Eq. ~9! that2¹p8 is not simply a pressure but a combination of¹ f anda force term2K(¹2e)¹e which is related to the surfactension given by Eq.~12!. The pressure inside the interfacdiffers from that outside so there is necessarily a pressgradient at the interface. The term given by Eq.~11! isneeded to balance the differential pressure otherwise theler or the Navier Stokes equation would require a changfluid velocity even in a stationary configuration, whichunphysical.

III. SOLUTION OF THE RELATIVISTICHYDRODYNAMIC EQUATIONS

The above hydrodynamic equations@Eq. ~9! or Eq. ~10!#can be solved after linearizing around the saddle configtion. The saddle point corresponds to the stationary soluwhene(r ,t)5e(r ) andv(r ,t)50 and alsoe satisfies@9#

2K~¹2e!1] f

]e50. ~13!

We can now write the equations of motion for small devtions about the stationary configuration by defininge5e(r )1n(r ,t) wheren is a small fluctuation in energy density anv501v(r ,t) and linearizing Eqs.~8! and ~10! around thisconfiguration:

] tn~r ,t !52¹•~vv~r ,t !!, ~14!

] t„vv~r ,t !…5¹e~2K¹21 f 9!n~r ,t !

1¹„~ 43 h1z!¹•v~r ,t !…. ~15!

Here f 95]2f /]e2, evaluated around the stationary configration. The dynamical prefactork is determined with theradial perturbations of the form

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n~r ,t !5n~r !ekt,

v~r ,t !5v~r !ekt. ~16!

It can be seen from Eqs.~14! and ~15! that the radial devia-tions are governed by the equations of motion of the folloing form:

kn~r !52¹•~vv~r !!,

kvv~r !5¹e~2K¹21 f 9!n~r !1¹„~ 43 h1z!¹•v~r !…. ~17!

Following Langer and Turski, we find the solution forn(r )in each of the following three regions:~i! the interior regionof hadron phase,r<R2j; ~ii ! the exterior region of QGPphase,r>R1j; and ~iii ! the interface region,R2j<r<R1j, whereR is the radius of the hadron bubble with originr 50. The interfacial region has a thickness of the orderthe correlation lengthj. Further, it is assumed that, everywhere outside the droplet, the energy densitye(r ) has thevalueeq , the quark density. Within the droplet,e(r ) is equalto the hadron densityeh . Thus, e(r ) describes a smoothinterfacial profile atr 5R going from eh to eq within a re-gion of roughly the order of the correlation lengthj. We thenevaluate the relative amplitudes in the above three regionmatching the values at the boundaries. Finally, we evaluakapplying the condition

E0

`

r 2n~r !dr50, ~18!

which is the conservation law implied by Eq.~14!.Before we proceed further, it may be noted here that

above set of linear equations@Eqs.~17!# is obtained from Eq.~10! which containsf (e) andK explicitly. As will be shownsubsequently, this form is suitable for the interfacial regiwhere¹e is nonzero. Such an equation has been used in@9# to evaluate the velocity profile at the surface regionorder to evaluate the energy loss due to dissipation. Inother approach, Ruggeri-Friedman@11# linearize Eq.~9! onlyin the exterior region with the assumptionp5cs

2e wherecs

is the velocity of sound in the medium. We will discuss tvalidity of the above assumption particularly near the stionary configuration as we proceed further. However,advantage of using Eq.~9! along with the above relation isthat one of the variables~sayp) can be eliminated so that thlinear equation becomes a simple wave equation. Therefwe adopt both approaches as in the following. We solverelativistic hydrodynamic equations by linearizing Eqs.~8!and ~9! in the interior and exterior regions whereas we uthe linear Eq.~17! in the interfacial region.

A. The interior and exterior region

In these regions,e(r ) is varying so slowly that the gradient energy can be ignored so that Eq.~11! is consistent with

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¹e] f

]e5¹ f→2¹p. ~19!

Since] f /]e50 at the saddle point@see Eq.~13!#, the aboverelation would implyp is independent of the energy densiat the stationary configuration. Like energy densityn, wealso consider a small fluctuation in pressure so that we hp(r )5 p1b(r ,t) @recall, e(r )5e(r )1n(r ,t)]. We assumethat the corresponding fluctuations satisfy the relationb5cs

2 n wherecs2 is a constant (cs could be the velocity of

sound in the medium around the saddle configuratio!.Therefore, in the interior and exterior regions, we solveequation

k2n~r !5cs2¹2n~r !1

k

vS 4

3h1z D¹2n~r !, ~20!

obtained from Eqs.~8! and ~9! after linearizing around thestationary configuration and also using the relation¹2b5cs

2¹2n. Such a relation has also been used in Ref.@11#, butwe differ in our interpretation of the pressure gradient.

Assuming spherically symmetric solutions of the form

n~r !5Constant

re6qr, ~21!

we get the relation

k25cs2q2, ~22!

where

k25k2

S 11~k/vcs2!S 4

3h1z D D . ~23!

The above relation holds both for QGP and the hadrongions, except for the fact that the viscosity coefficientsdifferent in two phases. The interior and exterior solutiotherefore, are

n~r !5A

rsinh~qr ! for 0<r<R2j ~24!

and

n~r !5B

re2q(r 2R) for r>R1j. ~25!

If k is small, the solution will be the one in whichn(r )varies slowly over a distance of the order of correlatilength j so thatqj!1. Sincek is related toq, next weproceed to estimate it by solving the linear hydrodynamequation in the interfacial region and matching it at tboundary.

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B. Interfacial region

As will be shown subsequently, the velocity varies asv}r 22 in this region so thatr 2 v remains constant. As a consequence,¹•v5r 22d(r 2v)/dr vanishes at the surface region. Therefore, ignoring the viscous term and eliminatingvfrom Eqs.~17!, an equation forn(r ) is obtained as

k2n~r !52¹•@¹e~2K¹21 f 9!n~r !#. ~26!

Further,k is assumed to be small so that in the first appromation we can completely neglect the terms containingk2.Thus, to a good approximation in the interfacial region,n(r )satisfies

¹•Fde

dr~2K¹21 f 9!n~r !G50. ~27!

We follow the procedure described in Ref.@6# to get thesolution of the above equation in the interfacial region~seeAppendix for detail!,

n~r !'2a~R!R2De

2sr

de

dr, ~28!

where De5eq2eh . This solution is quite similar to thafound in Ref.@6#, with n replaced bye anda which is now afunction of r evaluated atR. Note that in the quark and thhadron regions~far away from the surface region!, e(r ) isnearly constant. Thus, Eq.~27! becomes undefined in theseregions. Therefore, a different set of equations has beenin the exterior-interior regions as discussed in the previsections.

We can also get an expression for velocityv(r ) from therelation,

kn~r !51

r 2

d

dr@r 2vv~r !#, ~29!

Substitutingn(r ) from Eq. ~28!, we get

v~r !5D

r 2vE

0

r

rdrde

dr, ~30!

whereD is a constant. The above equation can be integrato give

v~r !'D

vq

R

r 2. ~31!

Recall that this result is consistent with our assumption tr 2 v is constant in the surface region.

C. Dynamical prefactor

For the interfacial region, the solution is given in Appedix. It remains now only to apply Eq.~18! to computeq ~ork). As in @6#, we can neglect the contribution coming frothe interior region (r ,R) and the terms of orderqR

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'Aj/R in the exterior region. The contribution coming frothe interfacial region is'2aR3(De)2/2s wherea is a func-tion of r related to the constantB and the second derivativof f with respect toeq . The exterior region contribution is'B/q2. Combining both terms, we get

q5S 2sB

a~R!R3~De!2D 1/2

. ~32!

Assuminga(R1j)5a(R2j)'a(R) and using the relation(]2f /]eq

2)21 for B/a @see Eq.~A7!#, we obtain

q5S 2s

R3~De!2

1

~]2f /]eq2!D 1/2

. ~33!

We can also eliminate]2f /]eq2 by using the relation

1

K

]2f

]eq2

51

jq2 , ~34!

where jq is the correlation length andK is related to thesurface tensions given by Eq.~12!. The choice ofK de-pends on the energy density profilee(r ). Following @9#, scan be related toK in the planar interface approximationTc as

s5K~De!2/6jq , ~35!

which will result in

q5Ajq /3R3. ~36!

Therefore, in the case of a nonviscous plasma, we get asimple relation fork given by

k5csAjq /3R35jq21Acs

2x3/35jq21 f ~x!, ~37!

wherex5jq /R. This can be viewed as the critical behaviof k that scales asjq

21f (x). However, this scaling law isdifferent from the dynamical scaling law that one finds in tcase of a nonrelativistic liquid-vapor transition@7,8# wherekscales asj0R23. While this needs further investigation, onreason for this discrepancy could be that, unlike the stscaling, the dynamical scaling depends on the dynamicalhavior of the system@16# which is definitely different de-pending on whether the medium is relativistic or nonrelatistic. The above result is also valid in the case of the viscplasma, only instead ofk, k will scale asjq

21f (x). There-fore, for viscous quark gluon plasma, this scaling resultsquadratic equation ink with solution given by

k5aq2

21csqA11a2q2/4 cs

2, ~38!

wherea5(4h/31z)/v. Since in the first approximationq}k, we can neglect the second term under the squarewhich is of higher order inq2 anda2 ~viscosity!. Finally, weget

01401

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k5q2

2vS 4

3h1j D1csq. ~39!

Using Eq.~36! for q, we can obtain a general expression fk for a viscous QGP as

k5csAjq /3R31jq

6R3

1

vqS 4

3hq1zqD . ~40!

Therefore, the prefactork can be written as the sum of twterms having a nonviscous (k0) and a viscous (kv) compo-nent. However, bothk0 andkv have simple dependence othe correlation lengthjq and the bubble radiusR. As can beseen from Eq.~40!, the first term is more dominating acompared to the second one particularly whenT is close toTc . However, as temperature decreases, the viscous cobution competes with that of the nonviscous one.

We can also express the above equation~40! in a differentway by assumingcs

2 as

cs25vq

]2f

]eq2

. ~41!

The above relation is analogous to the nonrelativistic expsion for velocity of sound in the medium which has a simirelation withv ande replaced byn ~density! @11#. Then fromEq. ~33! we get

q5cs21S 2s

R3

vq

~Dv!2D 1/2

, ~42!

where we have used the approximationDv'De since thepressure difference is negligible as compared to the difence in energy density. Now using the aboveq in Eq. ~39!,the prefactork can be written as

k5S 2s

R3

vq

~Dv!2D 1/2

11

cs2

s

R3~Dv!2 S 4

3hq1zqD . ~43!

As can be seen, the first term in the above equation issame as Eq.~4! as obtained by Ruggeri and Friedman corsponding to the case of a nonviscous plasma. The secterm is similar to the result obtained by Csernai and Kapuexcept with a minor difference, i.e., instead of 4, we havfactor ofcs

22 in the numerator@see Eq.~3!#. However this isa small difference which can be removed by redefiningK@see Eq.~35!#.

It may be mentioned here that the relationb5cs2n has

been used to obtaink as given by Eq.~40!. This is the mainresult of our work. It is also satisfying to note that we crecover the result of Csernai and Kapusta and of RuggeriFriedman under the assumption forcs

2 given by Eq.~41!. Inanalogy with the nonrelativistic case, we may interpretcs asthe velocity of sound around the saddle configuration~recallthat saddle point is the configuration where Eq.~13! is satis-fied!. Although the above interpretation needs further justcation, it is sufficient to say that the results of Eq.~43! can berecovered usingcs

2 as given by Eq.~41!.

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D. Result and discussion

We should point out here that there are several reasonassumptions that have been made in our derivation ofdynamical prefactor. As discussed in the text, most of thapproximations are the same as that used in the original wof Langer-Turski@6# since we use the same procedure excthat the equations follow relativistic hydrodynamics. An important aspect where we differ from both Refs.@9# and@6# isthe use of the relationb5cs

2n assumed to be valid in thquark and hadron regions (b andn are the radial deviationsof the pressure and energy density from the stationary stion!. Although, we differ in our interpretation, such a reltion has also been used by Ruggeri and Friedman@11# toeliminate one of the hydrodynamic variables. However,do not make any such assumption in the interfacial regAs a result, the linearized equation used in the interfaregion is different from the one used in the exterior-interregions. Within the above formalism, we have derivedexpression for the dynamical prefactork. Two important as-pects of our result are~a! the prefactork can be written as alinear sum of a non-viscous (k0) and a viscous (kv) compo-nent and~b! the non-viscous component (k0) which dependson two parametersR and j is finite in the limit of zeroviscosity. The present result onk0 is also in agreement withthe viewpoint of Ruggeri and Friedman that the viscositynot essential for the dynamical growth of the hadron bubbThis fact is also evident from Eq.~8! which implies a non-vanishing energy flowvv even in the absence of viscositOnly terms second order inv appear in the energy equatioin the presence of viscosity, and the momentum equacontains a term linear inv. This means that viscosity termare relatively unimportant in the energy transport for smvalue ofv. The momentum equation, however, indicates tviscosity influences the time evolution ofv. Thus, viscositycan serve to disrupt the energy flow and generate entropycannot be the only mechanism for energy removal.

The above aspect apparently is in contradiction togeneral expectation that transport coefficients like viscoand thermal conductivity are essential for the removal oflatent heat and hence for the growth of the hadron bub@7–10#. This point needs further clarification. The formatioof the hadron bubbles can be interpreted as the thermaltuation of the new phase within a correlated volume of radR and surface thicknessj. According to the fluctuation-dissipation theory, this fluctuation would mean certaamount of heat dissipation. We can understand the origithis heat dissipation as follows. In the absence of therconductivity, the dissipative losses that occur in a fluiddue to the coefficients of shear and bulk viscosity thatpend on the gradient and divergence of the velocity firespectively. In the case of an incompressible fluid,losses are due to the shear stress alone since the bulk viity that provides resistance to the expansion~or contraction!does not exist. However, due to the nucleation of the critsize hadron bubble, the pressure or the tension in the fluno longer uniform, the pressure inside the hadron bubbeing more than the pressure outside. Because of this psure difference, the hadron bubbles will keep expanding w

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lislees-h

a nonzero wall velocity. Thus, the fluid medium outside wexert a frictional force on the bubble wall~causing heat dis-sipation! whose magnitude depends on the pressure difence between the two phases@17,18#. In the field-theoreticallanguage, this dissipation corresponds to the coupling oforder parameterf to the fluid which acts as a heat batEstimates for it in the context of electroweak theory habeen given in Refs.@19,20#. Therefore, our nonviscous paof the prefactor corresponds to a dissipation of dynamnature which does not depend on any transport coefficielike viscosity or thermal conductivity. This dissipation bascally arises due to non-uniform pressure across the interfFollowing a different approach, Ignatius@21# had also de-rived k in the limit of zero fluid velocity to be'2/(hRc

2)whereh is a phenomenological friction parameter~not to beconfused with the shear viscosity of the plasma! responsiblefor the energy transportation between the order paramand the fluid. Recently, Alamoudiet al. @22# have also stud-ied the dynamical viscosity and the growth rate of the nucating bubble where the viscosity effects arise due tointeraction of the unstable coordinate with the stable flucttions. They estimate a growth rate which depends onR, j,and the self-couplingl „k'(A2/R)[120.003lTj(R/j)2] ….In the limit of weak coupling, the above growth rate scalesR21 which is also consistent with our result@23#.

We would like to mention here that we have also studthe homogeneous nucleation using the above prefactorfor nonviscous and viscous plasma@24#. It is found that thefirst order phase transition from QGP to hadron mattwhich proceeds through supercooling, generates additioentropy even when the plasma is nonviscous. The correcto the growth rate due to the viscosity is found to be smthough a large amount of entropy is generated due toslow evolution of the system.

Finally, we conclude this section with the comment thin case of relativistic heavy ion collisions, an appreciabamount of nucleation begins from a supercooled metastQGP phase at which the radius of the critical hadron bubis of the same order as the width of the bubble interface.such a point, the homogeneous nucleation theory may bdown. However, the system moves out of this problemaregion quickly due to the release of latent heat which heup the medium again towardsTc . In the other application, inthe case of cosmology, this problem is not serious, althothere the actual value of the dynamical growth rate may beless importance. In either case, this study has significancehomogeneous nucleation under the thin-walled bubbleproximation.

IV. CONCLUSION

To summarize, we have derived an expression fordynamical prefactor which governs the initial growth of crical size bubbles nucleated in first order phase transitionthe case of a nonviscous plasma, the dynamical growthis found to depend only on the correlation length and the sof the hadron bubble, which are two meaningful scalerameters to describe the critical phenomena at the transpoint. The correction to the dynamical prefactor due to v

2-6

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ctith

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an

icretiotin

iiea

sibuthma

thizerendheatanio

d-

-

by-

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-

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s.


cosity is found to be additive and does not affect the growprocess significantly though additional entropy is generadue to viscous heating of the medium. Since the prefadoes not vanish in the limit of zero viscosity, extra entropyproduced during the process of nucleation even whenfluid is nonviscous. Nearly similar conclusions are adrawn by Ruggeri and Friedman who had derived thenamical prefactor by solving relativistic hydrodynamics folowing a different approach. However, unlike their result, tpresent prefactor can be written as the sum of viscousnonviscous terms. Interestingly, using an assumption forlocity of sound in the medium~around the saddle configuration!, which has a form analogous to what is used for norelativistic plasma, the viscous and the nonviscous partsfound to be similiar to the results as obtained by CsernaiKapusta and by Ruggeri and Friedman respectively.

In the present work we solve relativistic hydrodynamequations both in the interior-exterior, i.e., quark-hadrongions and surface regions. The linear hydrodynamic equaused in the quark-hadron region is obtained after eliminaone of the variables using the relationb5cs

2n which is notvalid in the surface region. Therefore, a different equationused for the surface region which involves the extra gradenergy. This is where we differ from the Csernai and Kpusta method. Further, Csernai and Kapusta derivedk byequating the flow of the outward energy flux with the dispative loss due to viscosity of the medium, and the contrition due to a dynamical dissipation was not included. Onother hand, Ruggeri and Friedman solve the hydrodynaequation only in the quark region and use a set of boundconditions with certain assumptions. In this context,present formalism is more general as we solve the linearhydrodynamic equations in all space and obtain an expsion for the prefactor by matching the solutions at the bouary of the interface. Moreover, our result is different in tsense that it has a very simple dependence on the correllength and radius of the hadron bubble although the CKRF results can be obtained from it under certain assumpt

ACKNOWLEDGMENT

We thank A. Dumitru, W. A. Friedman, J. Ignatius, anD. K. Srivastava for many fruitful discussions and comments.

APPENDIX

We find the solution forn(r ) that satisfies the radial equation @see Eq. 27#,

d

dr F rde

dr S 2Kd2

dr21 f 9D x~r !G50, ~A1!

where n(r )5x(r )/r . Finally we solve forx(r ) from theequation

S 2Kd2

dr2 1]2f

]e2D x~r !5a~r !. ~A2!

01401

hd

orse

-

nde-

-red

-ng

snt-

--eicryeds--

iondn.

The above equation is quite identical to the one usedLanger and Turski@6# in the surface region. The only difference is that the constanta now depends onr as a(r )}(r¹e)21. ~Note that¹e peaks atr'R.! SinceR@j and¹e(r ) varies sharply in the rangeR2j<r<R1j, a(r )mostly depends on the¹e(r ) variation. Therefore, to a goodapproximation we can writea(r )}(R¹e)21. The generalform of the solution of Eq.~A2! is given by

x~r !5E dr8G~r ,r 8!a~r 8!, ~A3!

whereG is the Green’s function satisfying

S 2Kd2

dr2 1]2f

]e2D G~r ,r 8!5d~r 2r 8!. ~A4!

On either side of the interface]2f /]e2 is nearly constant.Using the relation¹2e50, it is easy to verify that

x~r !'a~r !S ]2f

]e2D 21

~A5!

is an approximate solution of Eq.~A2! at the interfaceboundary. Matching the solution in the interfacial regiogiven by Eq.~A5! with the solution in the interior regiongiven by Eq.~24! at R2j and with the solution in the exterior region Eq.~25! at R1j yield the following conditions:

A sinh~qhR!5a~R2j!S ]2f

]eh2D 21

~A6!

and

B5a~R1j!S ]2f

]eq2D 21

. ~A7!

Here the conditionqj!1 has been used. To get a solutioinside the interface, we follow the same procedure as thaRef. @6#, i.e., we use the spectral decomposition ofG as

G~r ,r 8!5(n

xn~r !xn~r 8!

ln

, ~A8!

where ln are thes wave eigenvalues andxn are the corre-sponding eigenfunctions. For values ofr nearR, the sum willbe dominated by the first term. This is becausel1.22K/R2 vanishes asR becomes large. Sincex1(r ).(K/s)1/2(de/dr) is sharply peaked at interface, using Eq~A3! and ~A8! we get

x~r !'2a~R!R2De

2s

de

dr, ~A9!

whereDe5eq2eh . We can also estimate the variation ofain the rangeR2j<r<R1j,

a~R1j!

a~R!'

¹e~R!

¹e~R1j!. ~A10!

Assumingj to be the half width of the full maxima, theabove ratio could be'A2.

2-7

r,

r-

ityore

f

s.

ter,

r-v. D


@1# J. Harris and B. Muller, Annu. Rev. Nucl. Part. Sci.46, 71~1996!.

@2# L.P. Csernai and J.I. Kapusta, Phys. Rev. Lett.69, 737~1992!.@3# P. Shukla, S.K. Gupta, and A.K. Mohanty, Phys. Rev. C59,

914 ~1999!; 62, 039901~E! ~2000!.@4# E.E. Zabrodin, L.V. Bravina, H. Stocker, and W. Greine

Phys. Rev. C59, 894 ~1999!.@5# J.S. Langer, Ann. Phys.~N.Y.! 41, 108~1967!; 54, 258~1969!;

Physica~Utrecht! 73, 61 ~1974!.@6# J.S. Langer and L.A. Turski, Phys. Rev. A8, 3230~1973!@7# L.A. Turski and J.S. Langer, Phys. Rev. A8, 3230~1980!.@8# K. Kawasaki, J. Stat. Phys.12, 5 ~1975!.@9# L.P. Csernai and J.I. Kapusta, Phys. Rev. D46, 1379~1992!.

@10# R. Venugopalan and A.P. Vischer, Phys. Rev. E49, 5849~1994!.

@11# Franco Ruggeri and William A. Friedman, Phys. Rev. D53,6543 ~1996!.

@12# We follow the procedure given in the first part of the LangeTurski formalism@6# which provides ak that does not dependon the thermal conductivity. However, thermal conductivwas included in the second part. As mentioned by the auththere is an error in the second part which they correct in R@7#. However, this error is unrelated to the first evaluation ok

01401

s,f.

in Ref. @6#.@13# L.D. Landau and E.M. Lifshitz,Fluid Mechanics~Pergamon,

London, 1959!.@14# S. Weinberg,Gravitation and Cosmology~Wiley, New York,

1972!.@15# P. Danielewicz and M. Gyulassy, Phys. Rev. D31, 53 ~1985!.@16# L.P. Kadanoff and J. Swift, Phys. Rev.166, 89 ~1968!.@17# M. Dine, R.G. Leigh, P. Huet, A. Linde, and D. Linde, Phy

Rev. D46, 550 ~1992!.@18# Guy D. Moore, J. High Energy Phys.03, 006 ~2000!.@19# S.Yu. Khlebnikov, Phys. Rev. D46, R3223~1992!.@20# P. Arnold, Phys. Rev. D48, 1539~1993!.@21# J. Ignatius, in Proceedings of Strong and Electroweak Mat

97, Eger, Hungary, 1997; hep-ph/9708383.@22# S. Alamoudi, D.G. Barci, D. Boyanovsky, C.A.A. de Ca

valho, E.S. Fraga, S.E. Joras, and F.I. Takakura, Phys. Re60, 125003~1999!.

@23# Due to large nucleation barrier atT5Tc , an appreciableamount of nucleation begins atT,Tc . In fact, most dominantfluctuations are those for whichR'j. Although our analysiswill break down at such a low limit, according to Eq.~37!, thiswould meank0}R21.

@24# P. Shukla, A.K. Mohanty, and S.K. Gupta, hep-ph/005219.

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Dynamical growth of the hadron bubbles during the quark-hadron phase transition