Bubble nucleation rates in first-order phase transitions
W. N. Cottingham Physics Department, University of Bristol, Bristol BS8 ITH, VK
D. Kalafatis and R. Vinh Mau Division de Physique Théorique*, IPN, 91406 Orsay Cedex and LPTPE, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France
Abstract This paper makes a critical examination of a formalism that is sometimes used for estimating bubble nucleation rates in a fluid undergoing a first order phase transition.
IPNO/TH 93-03 , Submitted to Physical Review B
* Unité de Recherche des Universités Paris 11 et Paris 6 Associée au CNRS y L Introduction
This paper deals with the rate of bubble nucleation in first order phase transitions. This subject is important, for example, in studies of the electroweak and Q.C.D. phase transitions of the early universe. These topics have recently received much attention but the developments presented in this paper are also relevant to phase transitions in general fluids encountered in other fields of physics and chemistry. More precisely we are concerned with the partition function Z and the associated free energy for a model in which the dynamics is that of a scalar field
Z can then be obtained from the functional integral [1]
Z = fd
c is a velocity, k and /3 = 1/ksT have their usual meanings. The functional integral
is over periodic functions: 0(r*,O) = ) will also depend on the temperature, and typical forms, appropriate for a first order phase transition are illustrated in Pig. 1. Over a range of temperatures / has two local minima, one at J is dominated by field configurations at and around Te this phase is metastable, the stable phase is characterized by field configurations ~ ^0- y In the metastable phase, below Tc, bubbles of the phase In this paper we will be concerned with transition temperatures that are high, in a sense that will be explained and initially we will consider /9 to be so small that we can restrict the integral to functions Of particular importance for computational purposes are the field configurations that give a stationary value to the exponent in (1.3). Such fields satisfy the equation -C2V2* + ^ = O (1.4) 4>{r) ~ (j)Q is one such field and fluctuations about this can be used to estimate F for the phase that is stable at low temperatures. Similarly, fluctuations about The latter can be achieved by cooling from temperatures T > Tc. Our treatment of the metastable phase is to expand f(4>) about <£ = 0 keeping only the terms quadratic in (j> to obtain the contribution y '/-•'* [*•«•«•*] (L5) fi2 = -f£(0) is positive (see Fig. 1). d I where (u;° )2 are the eigenvalues of the operator 2 2 2 ( M 0 = -C V + n (1.7) tu° has the dimensions of frequency. 3 ) -••*: If the product (1.6) contains high frequencies i»° > (fe/3) x the classical high tempera- ture expansion is invalid at these frequencies. Such modes should be treated quantum mechanically. We will return to this point later but in our approximation of quadratic expansions in Returning to the problem of the phase transition, an estimate of the critical bubble nucleation rate is given by the contributions to Z coming from fluctuations about solutions to (1.4) that describe the critical bubble profile [5]. These solutions have spherical symmetry [7] and the one centred at the origin of coordinates ^(f) = the fact that the phase is metastable outside the bubble requires the boundary condition This expression is obtained from eq. (1.3) by making the expansions = 4>b{r) + 64>{r) (1.10) + \{6 In (1.9) |V[^2] (1-12) ^ Terms linear in Sip have been discarded since they contribute nothing in an expansion about a stationary field configuration. Because of the boundary condition, V(r) = 0 outside the bubble at large r. A formal expansion for Zj, similar to that for Zo is 0S Zb = e~ {const) JJ { — } (1.14) n n but now w\ are the eigenvalues of the operator M = -C2V2 + il2 + V{r) (1.15) Adding Zt, to Z0 and taking the exponential suppression factor to be small we obtain to first order in e~&s F = F0 + Fb (1.16) The product in this expression can be seen to be the ratio of partition functions with and without the "potential" V(r). Comparing the operators MQ and M and since V(r) is negligible everywhere ex- cept within the critical bubble, it can be anticipated that for high frequency fluctuations w° —- —* 1 and the cut off dependence of (1.17) need not be severe. There are however Wn two important features that require careful attention. First it can be shown [7] that the. "breathing mode" of the bubble is unstable. This implies a negative eigenvalue w2 and hence W0 is imaginary. The inclusion of this eigenvalue in the product amounts to an analytic continuation of Z in the complex temperature plane with a branch point at the critical temperature Tc. F/, is now imaginary, and, it is this imaginary part [6] that can be interpreted as the bubble nucleation rate: f y/—w2 Im F for a single unstable mode. The second feature requiring attention are three zero eigenvalues, say w\ = tuf = w| = 0 corresponding to displacements of the bubble centre. The displacements for the three dimension translations give rise to the fluctuations «.»i(f) (1.18) I Because of "translational" symmetry ^j,(|f*+a|) has the same energy as ^»(|f|), and the V i three zero mode eigenfunctions ti(r) are J e,(rl = NdMr) (1.19) 5 "« •*, ._^_ N is a normalization factor which will be shown below to be * = Ts (1-20) In the evaluation of the functional integral (1.9) each factor — in (1.14) comes from an integral over qn For a translation a,- = JVg1-, hence the integrals over g,- are integrals over space and 1 (JL\3/zJLv where V is the volume. The infinity that results from the three zero eigenvalues can thus be identified with a volume factor. The normalization factor N can be calculated from an equipartition argument [7] which yields for each i. 2 2 3 c f{di By making a change of scale, and to lowest order in e j /(Ml + e)r))d3f = (1 - 3e) J f{Mr))d3f (1.24) and J(OiMO- + t)r))2d3r = (1 - t)J(diMr))2d3f (1.25) 2 2 [ c r 2 e r 3 zero first order in / [TT (9iM0 + ) )) + f(MO + ) ))] à *'™ t° « we have / [J(OiMr)H" + WMr))]*?= 0 (1.26) and, with (1.12), the result (1.20) follows, ' I The bubble nucleation rate per unit volume F = ~£y\/~uo lm ^ can ^e written from (1.17), and (1.22) as •><••• .JU. II. The evaluation of the prefactor The difficult task in the evaluation of the bubble nucleation rate F is to compute the prefactor (the factor multiplying e"^5). One purpose of this paper is to present a generalization of a theorem on the determinants of Sturm-Liouville operators which could be useful for the calculation of the infinite products in (1.27). The eigenfunctions of the operator M can be classified according to their "angular momentum" index /. For a given /, the eigenvalues of the radial operator 2 2 2 M1 = -c -^ + c ^^ + O + V(r) (2.1) or T are (21 + 1) fold degenerate. We also need to consider the operator a is a real number in the range 0 of Mia are denoted by eip(a,r) and its eigenvalues by wjp(a). The boundary conditions to be imposed on eip(a,r) are that eip(a,r) —> 0 for both r —> 0 and r —> R (where R is a large bounding radius). We define the infinite product A direct numerical evaluation of this infinite product is time consuming and we present a theorem which provides a simple indirect method of computation. To this end consider the differential equation ,r) = 0 (2.4) where r = O is a singular point. There is a regular solution yi(a,r) which behaves like r'+1 and an irregular solution yi(a,r) behaving like r~' at small r. Then we define ,:.( i ) The theorem, proved in the appendix, states that f,(a) = g,{a) (2.6) This is a generalization to three dimensions of the work of ref [8]. Hence the product of all the eigenvalues of a given / can be obtained by finding the solutions of just two differential equations, one with a = 0 the other with a = 1. For / = 0, /o(l) must be negative and hence So(I)- It is the square root of this term that gives the imaginary contribution to F. To obtain F from (1.27) the negative eigenvalue WQ has to be determined separately. For I = 1, /i(l) and hence Si(I), must be zero. The product of the eigenvalues excluding the zero mode can be obtained by replacing Mi with Mi + SSl2, with S being a small number and solve ^ £ hr) = 0 (2.7) If we denote by [J' the product excluding the zero eigenvalue, and define g[(l) to be g[(l) has the dimensions of time squared. From (1.27) we arrive at the approximate formula 0S 3li 1 p nf 1 afftt BS . in which we have assumed that the high frequency modes, which should be either removed or modified, make only a small contribution. III. High frequency modes I In this section we investigate the convergence of the product over / and the in- fluence of the high frequency modes for which the high temperature expansion is inap- propriate. We will consider / to be so large that the barrier factor —^-—- dominates \ the potential term V(r) so that V can be treated in first order perturbation theory. It is shown in the appendix that to first order / G,(r,r)V(r)dr (3.1) Ja Gi(r,r') is the Green's function for the operator Afjo of eq. (2.2). For large i Hence and the infinite product is linearly divergent. The source of the divergence lies in the high frequency modes. To show this, take the representation of the Green's function in the infinite volume limit, appropriate for large R ,2 f°° k2dk{TJi{kT)){T'ji{kT')) so that 0 t°° 1,2/M. tR )2V{T)r2dr (3.5) If we were to estimate the complete ratio of partition functions in perturbation theory we would write, to first order in V I - n [^i* -1 - i (3.6) As ^(2/ + l)[ji(kr)]2 = 1, (see for example [9]). f=o t -i with A=IJ" V(r)d3r. The integral over k is linearly divergent at large k and this corresponds to large frequencies. However, this is the perturbative expansion in the high temperature limit. To see how it must be modified if no high frequency cut off is applied, consider the dynamical equations appropriate to our system (1.1). ^ - C2V2 + fi2 + V(r)]* = 0 (3.9) If, for the moment, with frequencies I/Q — y/c2k2 + fl2. Treating V(r) in first order perturbation theory the frequencies are modified to 2 2 2 u = Jc Jt + il + i fv(T)d*T = u0 + -*- (3.11) y VJ UQV The ratio of quantum partition functions for each mode with and without V(r) is to first order: = 1 and for all modes 7.. t A*Û /JJiA 1 3 (2*) i/0 (3.12) \ This expression becomes identical to (3.7) in the high temperature limit. Even with no ; cut off the high frequency modes do converge when treated quantum mechanically. The divergent high temperature, or small h limit of (3.7) is of course equivalent to the product over partial waves and shows that if the high frequency cut off is beyond > the classical limit then the high frequencies must be quantized. The quantum statistical * < mechanics of first order perturbation theory is not computationally demanding so that ' one possibility is that the high temperature expansion is satisfactory for small I values and first order perturbation theory for the rest. Depending on the form of V(r) it would then be necessary that both approximations should be compatible over some 10 intermediate range of I. The fact that high frequency modes also contribute to small / need not be a serious problem since, in contrast to large I1 the integrals of (3.5) converge quickly for / small. IV. Discussion In this section we consider the situation in which one has a form, model or oth- erwise, for the Hamiltonian density and we give some criteria for assessing the utility of the suggested computational scheme. Given the temperature range of interest and the form for f( We consider first the situation of two phases in equilibrium described by P and $ are constants.The two minima of / are degenerate and the critical bubble radius is infinite. The interface between the two phases X0 = can be identified with the interface thickness, and the free energy per unit area T is "7* For the critical bubble to have a finite radius requires some deviation from equilibrium. A small modification of (4.1) that describes this is = *(|)2[d -|)2 -(I-)] 11 z is a temperature dependent parameter. With this form, fi2 is given by and our first criterion is that fc^T >> Ml. The potential (1.13) is given in terms of (j>b(r) by we envisage 0 and at the minimum, the excess pressure Po = —fmin is The critical bubble 3 2 S(R) = -^TtR P0 + 4*R T (4.8) the radius is determined by the condition that S should be stationary with respect to small changes hence S ~ "Mf - 3nR T (410) 12 We will take as a condition for the adequacy of a perturbative treatment to be that the angular barrier factor is greater than the maximum depth of the potential; from (4.6) 3P this maximum depth is —= so this condition is 9* This implies that only a few partial waves need a careful treatment if the bubble radius is not much greater than a few times x0, or that the degree of supercooling — should not be too small. We suggest therefore that a sound estimate of the prefactor can be made by the methods outlined here when there is a sufficient degree of supercooling. This is in fact a fortunate circumstance since the higher the degree of supercooling the smaller is the critical bubble size and the smaller the free energy barrier S. In practical calculations if /?5 is large, i.e. near the critical temperature, most effort should be put into the estimation of S, because it occurs in the exponent e~&s, and an accurate calculation of the prefactor is not imperative. If/95 is not large, however, precise bounds on the size of the prefactor become more urgent. The results of this paper suggest that as the prefactor becomes more important the easier it is to estimate. Two final remarks are in order that concern transitions which take place close to the limiting situations of no supercooling (z = 1 in the model) and such intense supercooling that one is close to the spinodal point (z = 0). First, for 2 = 1 the critical bubble size is large and although an accurate calculation of the prefactor is not so important, an estimate in terms of capillary waves on the bubble surface could well be more appropriate than the analysis suggested here. Clearly surface waves rather than bulk fluctuations capture more directly the essential physics. Second, at the spinodal point the transition faces no free energy barrier and there is no critical bubble that has to be created. There is much work in the literature on the evolution of spinodal transitions [10]. Very close to the spinodal point the critical bubble radius is very small and anharmonic fluctuations around the metastable phase become as important as the critical bubble nucleation rate. How close one has to be to the spinodal point before this situation occurs depends upon the actual temperature of the transition. In this situation a treatment in terms of an ( effective spinodal transition could then replace the analysis suggested here. à 13 X X Appendix Consider the operator ^ c2^±^ + a2+aV(r) (Al) c2, fi2 and a are positive numbers and 0 < a < 1. V(r) is a "potential" function. The eigenfunctions eip(a,r) of Mia satisfy the boundary conditions that they vanish +1 like r' at small r, and vanish at a radius r — R. Denote the eigenvalues of Mia by 2 w p(a), and the infinite product over all eigenvalues by ft(a), Also define gi(a)=yl{a,R)/yl{Q,R) (A3) where yi(a,R) is the regular solution of the differential equation Mlayi(a,r) = 0 (A4) with the boundary condition yi(a,r) —> r'+1 as r —> 0 and yi(a,r) —> 0 at r = Jî. To prove that fi(a) = gi(a), we first show that d/dalog(fi(a))=d/dalog(gt(a)). (Ab) The LHS of this equation can be computed from (A2) to be t% To find an expression for the derivatives of the eigenvalues, we differentiate with respect I to a the eigenvalue equations 2 Mlae,p(a,r) = w p(a)etp(a,r) (Al) 14 \ multiply by f/p(a,r) and integrate over r. This yields i *4W _ff 4 ,(a) da J0 In obtaining this last expression we have done some integration by parts and used properly normalized eigenfunctions. Sum over p gives ^log(/,(a)) = jT Gla(r,r)V(r)dr (A9) where Gia{r,T') is the inverse operator of Mj0 (Green's function) *'> („10) There is another representation of the Green's function that we also need and this involves an irregular solution of (Al), jf/(a,r) which is singular at the origin like 2/+îr~'- Since this condition alone does not fully define y we can impose at the far boundary yi(a, R) = 0. the Green's function has the representation Gia{r, r' ) =-iyi{a1 r)yt (a, r') r Turning to the RHS of (A5), d Differentiate (A4) with respect to a to obtain ,') [AU) ^ The general solution of this equation which is regular at the origin is t,r')dr' + Cyi(a,r) (AU) J with C constant. 15 ] Since ~jr[yi(a,r) = 1 independent of a, prr^jf1^ = 0 at r = 0, and C can be found C= f yi^,r')yi{a,T')V{v')dT' (415) and (A14) becomes 1 4«(«.*) = f Gta{r,r)V(r)dr (A16) yi(a,R) da J0 Hence d/da log(Ma)) =d/dalog(g,(a)) and since //(0) = 5/(0) = I1 one obtains f,{a)=g,(a) (418) To prove eq. (3.2) we integrate (A16) and use (All) with a = 0) so that to lowest order in a and g,(a) = 1 + J / w(0.')w(0,r)V(r)*r (420) Apart from normalization and a factor of r, y;(0, r) and jf/(O, r) are regular, and irregular spherical Bessel functions. Except near the critical temperature where the bubble sizes are large the integral (A20) can be anticipated to converge quickly within a bubble radius or so. Also, for / very large the spherical Bessel functions can be replaced by the first terms in their power series expansions [9] to give / For large enough /, V(r) will indeed be a small perturbation on the "angular momentum ; J barrier" —^—- and (A21) should be a good approximation to the asymptotic form • ' of gi(a) for large I. Ji 16 >• Acknowledgements This work greatly benefited from conversations with A. Comtet and R. Evans. References [1] J.I. Kapusta, Finite temperature field theory, Cambridge University Press, Chap- ter 1. [2] A. D. Linde, Nue. Phys. B216 (1983) 421. [3] A. D. Linde, Phys. Lett. BlOO (1981) 37. [4] S. Coleman, Phys. Rev. D15 (1977) 2929. [5] J. S. Langer, Ann. of Phys. 54 (1969) 258. [6] I. Affleck, Phys. Rev. Lett. 46 (1981) 388. [7] C. G. Callan and S. Coleman, Phys. Rev. D16 (1977) 1762. [8] S. Levit and U. Smilansky, Proc. Am. Math. Soc. 65 (1977) 299. [9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical FunctionslO.1.50, Dover (1965). [10] J. D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States, Springer-Verlag (1983) Vj 17 < JU Fig. 1 Forms for /(<]>) at three temperatures around a first order phase transition >