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 <f>{f, t) with Lagrange density Z can then be obtained from the functional integral [1] Z = fd<f> e-/>/"IiMB)'+**)1+**)] (1.2) c is a velocity, k and /3 = 1/ksT have their usual meanings. The functional integral is over periodic functions: 0(r*,O) = <j>{ftp) [2]. The free energy density /(^) which is usually obtained in realistic situations by performing an integral over all other fields but with <f>(r, r) held fixed. If <j> is a phenomenological field for which short wavelength (high frequency) fluctuations are unphysical, such fluctuating fields are not included in the integral. Also, even if 0 is a fundamental field short wavelength fluctuations about a fixed <j>(r,T) could have been taken into account in the construction of f(<f>), and again the integral (1.2) is restricted to those <f>{ftT) with a short wavelength cut off in their Fourier decomposition. / is a functional of <f> but in practice it is often taken in an approximate form as being a simple function of <p and this we do here. f(<j>) 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> — 0 and the other at <f> = fo(T). Above a critical temperature Te, <f> — 0 is the minimum and the free energy F, given by i Z = e-*F J is dominated by field configurations at and around <f> = 0. If the temperature falls below 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 <f> = <f>o can form and expand to bring the phase transition to completion. The rate of nucleation of bubbles that do so expand is an important parameter for characterizing the degree of metastability and an estimate of this rate can be obtained by a careful interpretation of the functional integral (1.2) [2]. 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 <j>(f) that are independent of r. In this approximation [3] z = This limit, independent of ft, is a classical thermodynamic limit. 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 <f>(r) ~ 0 give an estimate of F for the metastable phase. Here we are interested in low temperatures and the rate of nucleation of bubbles of the stable phase within the metastable phase. 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<p By expanding in normal modes [3], one obtains 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 <f> about stationary field configurations, a quantum treatment leads to the replacement 1 0h 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) = <j>b(r) satisfies -l^rMr) +^(Mr)) =0 (1.8) the fact that the phase is metastable outside the bubble requires the boundary condition <f>b (r) = 0 at large r, and to avoid a cusp at the bubble centre, -j— = 0 at r = 0. Formally, small fluctuations about this critical bubble profile give a contribution to Z of s 3 1 Zb = e-* fd[6ft e-'Z" ' ' [Wty+W'+nrW* ] (1.9) This expression is obtained from eq. (1.3) by making the expansions = 4>b{r) + 64>{r) (1.10) + \{6<t>{r)f ^iMr)) (LU) 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<l>b) d r = S (1.23) 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.