# Classical Becker-Dring cluster equations Rigorous results on metastability and long-time behaviour.

код для вставкиСкачатьAnn. Physik 2 (1993) 398-417 Annalen der Physik 0 Johann Ambrosius Barth 1993 Classical Becker-Doring cluster equations: Rigorous results on metastability and long-time behaviour Markus Kreer * Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, United Kingdom Received 16 February 1993, accepted 9 March 1993 Abstract. We consider the classical Becker-Ddring cluster equations with constant monomer concentration c, = z > 0 and c / = u,-,zc,-, -(b/+u/z)c,+b,+,c,+, , 122 as a model which describes the kinetics of a first-order phase transition. For a large class of positive coefficients u, and b, (including the ones commonly used in physics and chemistry) we prove the following: (i) When the monomer concentration z is slightly greater than zr = Iim,+- b/u, then all initial states - containing only subcritical clusters of size 1 < I (where 1* denotes the critical size of a nucleus and depends on the supersaturation z-z,>O) - converge within a fairly short time towards a metastable state. In this metastable state only subcritical clusters are present. The “metastable equilibrium” has an exponentially long lifetime TM exp ( C ( Z - Z , ) - ~(where ) C and w are some positive constants). - (ii) For times greater than the lifetime TM this metastable state breaks down in the following sense: as t-03 each of the c,(t) converges towards the Becker-Ddring steady-state solution f , ( z ) like c,(t)-fr(z) = 0 (exp (- 11,I t ) ) (where 1, < O is the eigenvalue closest to 0 of a certain infinite transition matrix) and the total mass of supercritical clusters (i.e. clusters of size I > I * ) diverges in this limit. For large times the cluster-number increases linearly in time in the sense that Emf-,- n ( t ) / t = J(z), where J(z)>O is the Becker-DLlnng steady-state current. For the average cluster size [= = ; Ic,(t)/ =; c / ( t ) ,we find for sufficiently large times algebraic growth in time t, that is, p, t1’(’-“)<7(t)< p 2 t 1 ’ ( 1 - n )(where O < a < l is the algebraic growth exponent of the u,- 1” and p , , pz are suitable positive constants). This bound covers previous suggestions due to computer simulations and heuristic calculations. c c , Keywords: Dynamics of first-order phase transition; Metastable states; Cluster growth; Classical nucleation theory; Spectral properties of infinite tridiagonal matrices. 1 Introduction The following investigation is a continuation of 0. Penrose’s work about metastable states in the Becker-Doring cluster equations [l]. We are concerned with the classical * This research was carried out under a Heriot-Watt University scholarship. M. Kreer, Classical Becker-Doring cluster equations 399 version, based on the ideas of Becker and Doring themselves [2], in which the concentration of monomers (one-particle clusters) is taken to be constant, that is, c1( t ) = z, while the overall density p can vary (one might therefore classify the model considered here as model A in the Hohenberg-Halperin scheme of dynamical phase transitions; see, for example [3]). In these cluster equations the system is modelled as a collection of clusters of one thermodynamic phase embedded in an otherwise uniform matrix. These clusters, of size 1 say, can change size through the gain or loss of just one monomer at a time. The Becker-Ddring cluster equations are kinetic equations describing the resulting changes in the cluster concentrations cl cl=Jl-,-Jl for 1 = 2 , 3,... (1.1) with the current J , = a l c l c l - b l + l c l , l for 1 = 1,2,. . . . (1 -2) As mentioned above we assume constant monomer concentration c1 = z. The transition rates al and bl are positive and correspond to birth- and death-rates in the theory of birth-death processes (eg. [4], [5]). The physical realization of this model is a system of clusters coupled to a monomer-reservoir. One might think of the distillation of malt whisky: the alcohol vapour (phase I), idealized as a one-particle cluster gas, is condensed in the “swan neck” of the distillation apparatus by the formation of large clusters. These large clusters, exceeding a critical size 1*, represent the liquid phase (phase 11) in this simple model. Although Becker and Ddring (and before them Volmer and Weber, and Farkas) introduced this model to describe nucleation phenomena and thus the kinetics of firstorder phase transitions, their treatment was based more on static than dynamical arguments ([2], [6], [7], [8]). In fact, Becker and Dtjring (1935) never stated the celebrated Becker-Daring equations explicitly in their famous paper 121. Their argument uses Einstein’s fluctuation theory together with Gibbs’ formula for the work necessary to create a critical droplet in a supersaturated vapour. Denoting byp the vapour pressure of the supersaturated vapour and by p m the critical vapour pressure they noticed that for small supersaturation p -pm>0 the probability of creating a critical droplet by fluctuations is for fixed temperature proportional to exp (- const/(p-p,)2) (with a well defined temperature dependent positive constant), and is therefore exponentially small as p -pm +O + . Apart from the unknown prefactor, which was calculated in [2] by kinetic arguments, the observation that the supersaturated vapour has an exponentially long lifetime was of major importance. The supersaturated vapour enjoys some stability and therefore Volmer and Weber use Ostwald’s terminology and speak about metastability [6]. In the following years much literature was devoted to formal calculations and improvements on the classical Becker-Ddring theory (e.g. [9], [lo], [ll], [12]) or numerical studies of the truncated system of equations derived from (1.1) - (i .2) (e-g. [131, 1141). One important new approach has been developed by Binder, in collaboration with Stauffer, Miiller-Krumbhaar, and others (e.g. [15], [16]; for a review and more references see, for example [17]). Starting from a master-equation description of dense gases, binary mixtures, Glauber kinetic Ising models etc., theories for nucleation, coagulation and droplet growth are derived by reformulating the dynamics in terms of “clusters”. Neglecting nonlinear coagulation terms in these cluster theories leads to equations, Ann. Physik 2 (1993) 400 which are formally the same as the Becker-Doring equations (1.1)- (1.2). Examples of utilizing these equations to evaluate data of computer simulations can be found e.g. in 1161, [191. Just recently it was proven rigorously in [I] that for suitable hypothesis on the transition rates aI and bl the infinite system of first-order differential equations (1 .l) - (1.2) possesses a metastable solution for z-z,>O small, where z, denotes the critical monomer concentration (notice that the monomer concentrations z and zsrespectively correspond to the vapour pressures p and p , respectively of Becker and Doring). However, some questions concerning both uniqueness and stability against suitable perturbations of the metastable state remainded open. We shall show here that a large class of physical initial states converges rather fast towards the metastable state and hence answer these questions positively. We shall consider the case in which the supersaturation of monomers z-z,>O is small, that is physically speaking, we shall discuss the physical situation near the coexistence curve. Finally we remark that instead of keeping the monomer concentration c1 = z constant one can modify the classical Becker-Doring equations by requiring conservation of density, which leads to an additional differential equation for c1 and makes the system of differential equations nonlinear. These equations have been of interest for the understanding of computational studies of the Kawasaki kinetic Ising model (e.g. [22]) and a corresponding truncated version has been studied numerically in [23]. For a rigorous discussion of these nonlinear equations we mention [24], where existence and equilibration of a unique positive solution is proven, and [l], where existence of a metastable solution is proven. The outline of this paper is as follows. In Chapter 2 we introduce Hilbert space methods to obtain some preliminary results. The methods developed here are also of some relevance in the theory of birth-death processes [25]. In Chapter 3 we show that all initial cluster distributions that contain only subcritical clusters approach in a “short” relaxationtime a metastable (one-phase) state, whose lifetime is much larger than the relaxationtime. Chapter 4 is concerned with long-time behaviour and gives especially an asymptotic bound on the mean cluster size which is algebraic in the time t. Chapter 5 contains a final discussion and conclusion. 2 Preliminaries Throughout the paper we shall assume that the transition rates a/, bl satisfy the following hypothesis (cf. [l]) for all positive integers i e N: (i) O < A ‘ i a < a l < A i a , O<a<l .f --1 ati-1 = (11) o ( fI-) a1 bl+l b (iii) -s--[ al,l al bl+l --1 bl =O(P) bl = Z,>O lim I-+- (iv) 3G,G’,y,y’>O aI with O < y < l : b z,expGi-Y<--’<z,expG’i-Y’ a1 . M. Kreer, Classical Becker-Doring cluster equations 40 1 The usual choice of the transition rates for the classical Becker-Doring theory of nucleation (see, for example [8],[26]) al f 2 1 3 , bl a1exp GI - ' I 3 satisfy our hypothesis. The simplest solution of (1.1)-(1.2) is the equilibrium solution determined by the condition of vanishing currents JI for f = 1,2, . . .: - - where For supercritical saturation z > z, it follows that the equilibrium cluster distribution (2.1) is decreasing up to a minimum I* Q , z ' L Q / + ~ z ~ +I '= 1, .. .,f*-I (2.3) where the critical clustersize I* is defined uniquely by with the definition bI = co for convenience. For f+ 03 this equilibrium solution has the asymptotic behaviour cLBD)- const (z/z,>',that is, tends to infinity like an exponential. Indeed, this equilibrium state is unphysical and as demonstrated later the solutions of the Becker-Doring equations tend to a bounded steady-state solution. One is therefore not forced to remove clusters exceeding a critical size (see, for example [8]). In the following two lemmata, proven in [l], we summarize results we shall need later. The first lemma gives information about the steady-state solution in the supercritical case. Lemma I For each z > z, the second-order difference equation with boundary conditionsfI(z)= z and liml.+afr(z)finite has a unique positive solution m with the steady-state current 402 Ann. Physik 2 (1993) Further ( i )forfixed z > z s : a l + t f r + l ( z ) s a l f ( z )1 = 1 , 2 , . . . (ii) for fixed I : fr(z)sfi(z')if and only i f z s z ' / \I The second lemma is a modification of some crucial bounds due to [I]: Lemma I1 For z >z,, A 2 1 fixed and initial data satisfying with finite second moment c l2C/(0)< m a(0) = (2.9) m /= 1 the Becker-Doring equations (1.1) - (1-2) have a unique non-negative solution satisfying O S c / ( t ) S A f i ( Z ), t r O , l = 2 , 3 , . . . (2.10) and c 12cl(t)sa(0)exp ( 4 A A z t ) c W O s a ( t )= co , tz0 . (2.11) I= 1 With the definition of I* as in (2.4) we have for the following quantities (2.12) the bounds Mo(t),<Mo(O)+J*t (0)+ to J * + to J* (2.13) ('ul)'-' - ?):+?+I( where (2.14) M. Kreer, Classical Becker-Doring cluster equations 403 1*+1 to = Az (2.15) fl=- 2 - a (2.16) l-a J* = Aai.QI.zP + l . (2.17) Further define E = (z-z,)/z, >0. Then there exists E~ >0 such that the following is true ( i ) I* is at most algebraically large as &-+O+, i.e. for E E (0, e0) the following bound holds -l/y' . O<I*<(-&&) (2.18) ( i i ) For A 2 1 fixed, J* is exponentially small as & -' Of, ing bound holds i.e. for E E ( 0 , ~ ~the ) follow- (2.19) Prooj All the arguments used in [l] work with the additional factor A r 1 to prove this modified Lemma I1 in our case. Let us discuss now the infinite system of linear differential equations (1.1)- (1.2) from a different point of view. Therefore let us introduce the following change of variable: xl = dlcr,l for 1 = 1,2,. . . (2.20) where for 1=1,2, ... . (2.21) - fa Notice that d f as I+ for z>z,. The advantage of this transformation (2.20) is that it makes the right-hand side of the Becker-Doring equations (1.1) symmetric in the sense that the corresponding transition matrix is symmetric: k1= a l z 2 - ~ 1 x 1 + r l l x 2I ,= 1 (2.22) if= ? f , - l X f - , - 61Xf+Y/[Xf+1, I = 2,3, . . . (2.23) with the transformed initial data xf(0)= d i c ~ +(0), l I = 1, . . . and (2.24) (2.25) 404 Ann. Physik 2 (1993) Using a more compact notation, (2.22)-(2.23) can be written as i=Ax+b (2.26) where x(0)= (xl(0),x2(0),. . .)' are the given initial values, x = x ( t ) = (xl(t),x2(t), . . .)T and b = (a,z2,0,0,. . .)', and finally the infinite matrix operator A with the representation This formal equation (2.26) with the infinite matrix representation (2.27) makes sense within the Hilbert space H = l2(C), where we denote the canonical norm as usual by 11 . 11 2. Obviously x E H (i.e. llxll 2 < 03) if and only if Notice that by this condition the Becker-Doring equiiibrium given in (2.1) does not correspond to an element in this Hilbert space H for z > z , since dlcjfy)- ( ~ / z ~ ) " ~ + o l as l+m. Our first theorem contains useful properties of the infinite matrix operator A defined on some appropriate domain in H. It also provides a unique solution for the initial value problem (2.26) in the Hilbert space by means of analytic semigroup theory (cf. [27]) and will be used for the discussion of the long-time behaviour of solutions. Theorem I11 The linear operator A: 9 ( A ) + H = 12(C)with domain where for convenience we set ~0 = 0, defined by the infinite matrix representation (2.27) has the following properties: ( i ) A is closed and symmetric. (ii) A is negative semidefinite and 0 is not an eigenvalue of A . (iii) A has a compact resolvent (A- c)-* for some c>O. Moreover A is the self-adjoint generator of an analytic semigroup (2.28) and the initial value problem (2.26) is solved uniquely for x(0)E H by X(t) = S(t)(x(O)-Xeq)+Xeq (2.29) M. Kreer, Classical Becker-Donng cluster equations 405 where for z>z, the element xeq= (d,f z ( z ) ,d2f3(z),. . .) E H is the unique solution of - [ is compact and b E H. Finally we have the following estimate in r2(e) 0 = A x + b in H since A (2.30) where A l< 0 is the maximum negative eigenvalue of A . ProoJ Some easy computations using A , = A If@, that is, A restricted to the dense set of finite sequences& = ( x e H = 12(C): 3moE iN such that x,,,= Ovm>m,] c Q ( A ) , show that and for (ul,. . .,uN,O,. . .) E f o = ~ ( A o ) since by definition u,+~ = 0, so that A . is negative semidefinite. It is easy to check that A is closed and hence a closed extension of A,: Take a sequence ( x ( ~ ) )C, ~9~( A )such 12 12 that xfn)-+xE H and Ax(")-+yE H as n-r 0 3 . We have obviously, for each I E iN that is, A x = y and x E 9 ( A )and hence A is closed. Since A is a closure of A,, closing the negative quadratic form associated with A . (see, for example [28]) leads to the desired properties and Notice that &,Ax) = 0 implies x = 0, i.e. 0 is not an eigenvalue of A . Thus we have shown (i) and (ii). To shown (iii), namely that the resolvent R c ( A )= (A- YO)-' is compact we consider for C>O sufficiently large A-CU = (O+F)D where D = diag {- (6,+ 0) and the operator F has the infinite matrix representation 406 Ann. Physik 2 (1993) - F= L 0 0 ... 0 0 ... 0 0 ... O ... 4+C This linear operator F is a bounded operator on H, as for sufficiently large C>O its operatornorm can be estimated by 1 IF(I < 1 because the following inequalities hold for all I = 1,2,. . . o<-c-t t l 4+1+C 1 2 Since the positivity is trivial for C > O we have to prove the right-hand member of the inequalities, that is, e.g. for the first pair of inequalities using (2.24) and (2.25) This is equivalent to The right-hand side here tends to 0 as I-+= by hypothesis and is thus bounded above. Therefore it is sufficient to choose C greater than or equal to this upper bound of the right-hand side. The second pair of inequalities holds by the same argument. Finally choose C: greater than the maximum of these two upper bounds. Thus we have shown that llFll 2 < 1, and therefore (I+F)-' is a bounded linear operator on H by the convergent von Neumann expansion. Further D is compact (because c > O and 6,+ C- 03 for I- 03; see, for example [30]) and so is the product of a compact and a bounded operator -' D-'(O+F)-' = (A-CU)-l = R c ( A ) . Since A : 9 ( A)+H is a negative semidefinite, closed symmetric operator with a compact resolvent & ( A ) for a c > O sufficiently large we conclude that the deficiency index is (0,O) and thusA is self-adjoint ([29], Chap. V, 0 3 Sect. 4, p. 271). Its spectrum is pure- 407 M. Kreer, Classical Becker-Doring cluster equations ly discrete consisting entirely of isolated eigenvalues ([29], Chap. 111, $ 6 Sect. 8, p. 187) and for the unique analytic semigroup generated by A ([27]) s ( t ) = e A r , t>O the following bound holds where 1,< 0 is the maximum eigenvalue of A closest to 0. Finally we notice that for z > z , the sequence (fi(z))reMis bounded and since Qlz'- (5)' as I-. 03 we see that by (2.20) and (2.21) indeed xeqE H . By construction as the \GI steady-state solution is satisfies 0 = A xeq+ b. Since A is compact and b E H we conclude further that xeq= -A - ' b is the unique solution. It is also easy to verify that (2.29) solves the initial value problem (2.26); by standard arguments (e.g. [27]) it is the only solution. This concludes the proof. Q.E.D. -' 3 Metastability The following theorem provides some crucial bounds which describe the timescale of the relaxation into the metastable state and the timescale of the lifetime of this metastable state. The metastable state considered here is characterized by the presence of subcritical clusters ( I < / * ) only - the mass of supercritical clusters (1>1*) is vanishingly small. As the coexistence curve is approached, i.e. z-+z,, the lifetime of the metastable state becomes much larger than the relaxation time into this metastable state. The idea behind the proof is a perturbation argument in conjunction with an estimate on a certain eigenvalue. The additional condition imposed on the transition rates a/ and bl is not too restrictive as it is satisfied by all the ones used in literature (see, for example [8], [26]). Theorem IV Let there be some positive constants K and v such that the transition rates a, and bl satisfy in addition to hypothesis (i)-(iv) the following condition --->bl-t bl -K , a1 1' I=2, ... . Let E = (z- z,)/z, > 0 with 0 C E < 1 and A 1 1 be a positive number. Consider any initial configuration satisfying with l s l o s l * ( ~ ) . Then there exists a unique, non-negative solution of the Becker-Dbring cluster equations (1.1)-(1.2) for which the following is true: 408 Ann. Physik 2 (1993) ( i ) There exist positive constants E ~ KR, , Cg, xg such that for -(l'y)-- v'y' the following bound holds KR E ' E E(0,~~ and ) t> T R = where m = m a (v,1 /2] and y defined in hypothesis (iv)satisfies 0 < y < 1. (ii) There exist positive constants eo, KM, r, C,, xisuch that for E E (0, E ~ and ) t IT, = KMexp [.(:)("')-'] the first moment M l ( t ) , i.e. the mass of supercritical clusters, is bounded as follows Proofi Existence, uniqueness and non-negativity follow from Lemma 11. The condition (3.2) guarantees further that all the important bounds from Lemma I1 can be applied here. Let us define now the following tridiagonal matrix and the following quantities (3.6) (3.7) E(t) = (CZ(t),. . .,C" ( t ) ) - (3.8) Then we can write the Becker-Doring cluster equations as ?(t) = A E ( t ) + 5 + i i ( t ) (3.9) We notice that 7 = (f2 ( z ) ,. . . , f r o (2)) is the unique solution of 0 = A^?+ 8. We shall now discuss the Eqs. (3.9) treating 2 ( t ) (which includes information of the higher cluster concentrations cl*+l(t),cl.+2(t), . . .) as a perturbation. Define the dia- l%,. .,g] gonal matrix Q = diag - . Qii(t) and A = Q A Q - to derive from (3.9) and set X(t) = QC(t), 6 = e6, R(t)= 409 M. Kreer, Classical Becker-Ddring cluster equations k(t) = AX(t)+5+7?(t) . (3.1 1) Writing Teq= Qx we can express the solution to (3.11) as follows, using the fact that A is a symmetric matrix (see Lemma VI in Appendix) (3.12) Now notice that by Lemma VI in the Appendix A is symmetric and thus (3.13) because all eigenvalues In,n = 1,2, .. . I*- 1, satisfy (3.14) Together with the bound (2.18) on I* we find for E E [O,E,] (3.15) Further using (3.2) and Lemma II we obtain and in the same way (3.17) With these bounds (3.13), (3.16) and (3.17) we derive from (3.12) the following estimate (3.18) We choose now some positive number KR and define for E E (O,E,,] 410 Ann. Physik 2 (1993) and set further m = min.I+,.{ Using the bounds on I* and QI*z" from Lemma I1 (namely (2.18) and (2.19) because QI*z'*cJ*),and the bound on ,I* (3.15) we conclude from (3.18) that some positive constants C, and xg can be choosen such that for t > TR the following bound holds (3.20) By the hypothesis on the transition rates al and bl we have further Q2z2rQ1z'for I = 2,3, . . ., I * ( & ) and thus from f ( t ) = &(I), where the diagonal matrix 0 is defined above, it follows that and thus (i) is proven by combining (3.20) and (3.21). The bound in (ii) is a direct consequence of the result in [I] given in Lemma 11. Choose some positive numbers KIMand r such that T, = K,e Hence for TM(E) O<r<iL G'lY. Define for E Pi-Y E (0, &o) j-& - ( I - Y ) / Y E E(0,~~ we ) (3.22) find from the bound on MI( t ) given in (2.14) for t i TM= 2141-a) OSM1(t)5Az where the positive constants are defined as (3.23) (3.24) and the proof is finished. Q.E.D. M. Kreer, Classical Becker-Doring cluster equations 41 1 4 Long-time behaviour and cluster growth The last theorem deals with the long-time behaviour and the average cluster growth for large times, after the break-down of the metastable state. Theorem V Let the transition rates al and bl satisfy the hypothesis ( i ) - ( i v ) from above. Let the initial conditions be such that their second moment isfinite and that for some A 2 1 the following bound holds 0 Icl(0)~ A f r ( z ) ,1 = 2,3, . . . . Then the solutions cr(t)of the classical Becker-Doring cluster equations (1.2)- (2.2) have the following properties: ( i ) The solution is non-negative and unique. Each of the q ( t ) converges towards its steady-state value like an exponential where ,I1 < 0 is the maximum eigenvalue of the operator A from Theorem 111 and Kl = r/ C p"= 2 I ci(O)-fr(z)I 2 / Q ~ ~ ' (ii) There exist a positive number T and positive constants 0 < K , 5 K~ such that for t>T the following bound on the total number of clusters per unit volume n ( t )= C p"= cl(t) holds VEZZ And in the limit t+a, lim n ( t )= J ( z ) t t+o (4.3) where J ( z )> 0 is the steady-state current from Lemma I, Eq. (2.7). (iii) Define the average cluster size i ( t ) as Then there exists a positive number T and there are constants Ocpl 5 p 2 such that for t > T the following bound is valid Proox The decay property (4.1) follows immediately from Theorem I11 and using Lemma I1 finishes the proof of the statement (i). It remains to show (ii) and (5). First of ail notice that the initial conditions have finite second moment and by (2.1 1) from Lemma I1 the following bound on cr(t)is obtained 412 Ann. Physik 2 (1993) 0 5 q ( t )Ia(0) exp (4AAzt) i2 Hence by the hypothesis on the transition rates (i)-(iv) there exists a continuous function K ( t ) increasing in t such that Thus we obtain after summation of the integrated version of the Becker-Doring equations where we have used Weierstrass convergence argument to perform the last summation. For the proof of (ii) we deduce from this last identity with help of Lemma I that t - n ( t ) = n(O)+Sd7(J(Z)+bz[fi(z)-cz(r)l) 0 With (4.1) we derive the following inequality and from this =,i For (iii) we argue in the following way: For sufficiently large t> 0 choose the integer iM(t)as large as possible such that M. Kreer. Classical Becker-Doring cluster eauations 41 3 since all terms are non-negative in the last twosums. From this we deduce with regard to (4.6) c c/(t)2iM W f(t) I= 1 m c 1, 1.M cl(t)fA (z-ZM)h(z)ZA lfi(z) I= 1 I- 1 I= 1 and thus ( zz) holds ~ (i.e. f i ( z )- l/al - I-"), Since by Lemma I the inequality ( J ( z ) / z ) ~ a L f r al we conclude that there exists F> 0 (due to the properties of the Riemann integral) such that c and from (ii) /cl f I p"= cl(t)I~ ~ that t ,we might choose by (4.6) the integer I, at least as large as (4.8) As before we conclude that there exists some F > 0 such that Using this bound together with (4.8) we obtain from (4.7) that there exists some positive constant T and some ,ul> O such that the following bound holds for t> T The bound p ( t ) / n ( t ) ~ p ~ f l ' ( ~for - " )large times follows from (2.14) in Lemma I1 since p ( t ) = icl(t)+Ml(t)~A If,(z)+M,(t) and (4.2). This concludes (iii). (Q.E.D.) c 5 Discussion Following the custom of identifying the subcritical clusters (is/*) with phase I and the supercritical clusters (Z>l*) with phase I1 (e.g. [l], [16]) we notice that Theorems IV and V are in agreement with the description of metastability given by Penrose and Lebowitz ([20], [21]): 414 Ann. Phvsik 2 (1993) (i) Only the thermodynamic phase is present. (ii) The metastable state has a very long lifetime. (iii) Once the system has left the metastable state, it is very unlikely to return. The first and second statements follow from Theorem IV which shows that for exponentially large times t ITM(&) the supercritical clusters (phase 11) give only an exponentially small contribution to the total system; the subcritical clusters (phase I), however, are after an initial relaxation process almost in equilibrium. The third statement follows from Theorem V since after a sufficiently large time the contribution of the supercritical clusters will never become exponentially small again. Furthermore Theorem IV states that any intial configuration containing only clusters of subcritical size approaches its “metastable equilibrium” rather fast compared with the lifetime of this metastable state. Indeed, the time scales for this equilibration towards the metastable state TR(&)and for the lifetime of the metastable state T’(E) (where in both cases E = (z-zs)/zsdenotes the supersaturation and thus the distance from the coexistence curve) can be compared by means of (3.19) and (3.22). As E+O we observe the following so that as the supersaturation goes to 0 the rate of decay of the metastable state is exponentially small compared to the rate of equilibration. Thus we can relate Theorem IV to the observations in [13] and [14], that in the numerical solution of the truncated Becker-Doring equations (1 .l) - (1.2) for small supersaturation the small clusters (up to the critical size [*) reach their steady-state concentration very rapidly while the large clusters are far away from their steady-state concentration. We emphasize also, that our rigorous estimate on the decay-rate A* in Eq. (3.15) can be related to the formal expressions of the time-lag for the approach to the steady-state in the truncated Becker-Doring equations (e.g. [12], [is], [261). Our result in Theorem IV also implies some form of uniqueness and asymptotic stability of the metastable state if we do not distinguish metastable states which are exponentially close as the supersaturation 6-0 +. Theorem V concerning the long-time behaviour of the Becker-Doring equations explains the numerical results in [13] and [14], where it was found that the numerical solution converges towards the bounded Becker-Doring steady-state solution (2.6) rather than the unbounded equilibrium (2.1). Theorem V also states that the growth of the average cluster size T is algebraic in t as t+ 00. We might compare this with [12], where a heuristic argument is presented that the time needed for an average cluster to grow from some cluster size above the critical size I* to some large size 7sI * is for the BeckerDoring theory (in d = 3 dimensions) approximately like t - 71’3. There a continuum approximation for a truncated system of Becker-Daring equations was used. Since for a sufficiently large time the contribution of the large clusters ( bI*) will dominate (as M,(t)+ao for t+a) we can compare the estimate of [12] with our rigorous bound (4.5) on the average cluster growth; taking the classical Becker-Doring value for the exponent a = (d- l)/d ([8], [26]) with d = 3 we find agreement. Finally we want to emphasize that the Becker-Doring model has been used to understand the dynamical properties of the Glauber kinetic Ising model. In this context 415 M. Kreer, Classical Becker-Dbring cluster equations reference [ 181 is of interest, where the classical Becker-Doring theory of nucleation was applied to explain heuristically the results of Monte Carlo simulations on the 3-d Ising model with Glauber dynamics: starting with a pure phase (all spins down) in a weak external magnetic field oriented in the other direction the long-lived metastable equilibrium is established after just a few MC-steps per spin. Furthermore the droplet growth law in the late stages found in [18] d P3= const, - c c o n ~ t ~ ~ - " ~ dt - is consistent with our rigorous bound (4.5) for a = ( d - l ) / d with d = 3. Just recently R. Schonman and coworkers [31] have been able to explain this observed metastable behaviour in the Glauber kinetic Ising model by means of rigorous arguments starting from first principles in statistical physics (the finite-dimensional version of this work is the well known CCO theory [32]). Even though one does not expect too high accuracy in the description of first-order phase transition by means of the model discussed here, the model nevertheless describes qualitatively a typical scenario for a first-order phase transition. We have proven rigorously that in this model the essential features such as relaxation towards a longlived metastable state and algebraic cluster growth for large times are present. Thus we have good reason to hope that more general cluster theories, allowing coagulation and fragmentation processes (e.g. [ 16]),will provide a further understanding of the dynamics of first-order phase transitions, but we shall not enter into this discussion here. A Appendix Lemma VI Given the tridiagonal matrix A E IR"-l)x(N-l) defined as follows B = diag (bz, . . .,b ~ . ] Then ( i ) A,QA'Q-' - - - and BAB-' have the same eigenvafues. (ii) A = Q A Q is tridiagonal symmetric matrix and has therefore only real eigenvalues. (iii) The tridiagonal matrix -' 416 Ann. Physik 2 (1993) has the following bound on its real eigenvalues A,,, n = 1, . . .,N1 with the definition bl = non-positive. +m. Under the hypothesis -2bl-1 bl a1-1 a1 all the eigenvalues are ProoJ (i) and (ii) are obvious. For a matrix F = (Fii)i,jE RMxM GerSgoryn's theorem states that all eigenvalues are within the union of discs UK E C: I c-Fii\ I C $ IFv/],where the prime indicates that the summation is over all j # i. If the matrix F has only real eigenvalues A,,we obtain the following bound ltc - - - -' Using the matrix F = B A B from (iii), where M = N- 1, we obtain the desired estiQ.E.D. mate after some obvious index-shift and the proof is finished. It is a pleasure to thank Professor 0. Penrose (Edinburgh) for helpful advice and stimulating discus- sions and Professor K. Binder (Mainz) for his hospitality and enlightening discussions on the subject during a visit in Mainz. References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] 0. Penrose, Commun. Math. Phys. 124 (1989) 515 R. Becker, W. Doring, Ann. Physik 24 (1935) 719 P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435-479 W. Ledermann, G.E.H.Reuter, Phil. Trans. Roy. SOC.London Ser. A246 (1954) 321 N. G. van Kampen, Stochastic processes in physics and chemistry, North-Holland, Amsterdam 1981 M. Volmer, A. Weber, 2. Phys. Chem. 119 (1926) 277 L. Farkas, Z. Phys. Chem. (Leipig) A125 (1927) 236 R. Becker, Theorie der Wame, 2nd ed., revised by W. Ludwig, Springer-Verlag, BerlinHeidelberg-New York 1978 J. Frenkel, Kinetic theory of liquids, Oxford University Press, London 1946 J.B. Zeldovich, Acta Physicochim., URSS, 18 (1943) 1 F. Kuhrt, Z. Physik 131 (1951) 205 M. Kreer, Classical Becker-Doring cluster equations 417 J. Feder, K.C. Russell, J. Lothe, G.M. Pound, Advan. Phys. 15 (1966) 1 1 1 W.G. Courtney, J. Chem. Phys. 36 (1962) 2009 F.F. Abraham, J. Chem. Phys. 51 (1969) 1632-1638 K. Binder, D. Stauffer, Advan. Phys. 25 (1976) 343 K. Binder, H. Muller-Krumbhaar, Phys. Rev. B9 (1974) 2328 J. D. Gunton, M. San Miguel, P. S. Sahni, The dynamics of first-order phase transitions, in: Phase Transitions and Critical Phenomena VIII, C. Domb, J. L. Lebowitz (Eds.) Academic Press, London 1983 [18] D. W. Heermann, A. Coniglio, W. Klein, D. Stauffer, J. Stat. Phys. 36 (1984) 447 [19] D. Stauffer, Ann. Physik 1 (1992) 56 [20] 0. Penrose, J.L. Lebowitz, J. Stat. Phys. 3 (1971) 211 [21J 0. Penrose, J. L. Lebowitz, Towards a rigorous molecular theory of metastability, Studies, in: Statistical Mechanics VII (“FluctuationPhenomena”), E. W. Montroll, J. L. Lebowitz (Eds.), NorthHolland, Amsterdam 1976, 1987 [22] 0. Penrose, J.L. Lebowitz, J. Marro, M.H. Kalos, A. Sur. J. Stat. Phys. 19 (1978) 243 [23] 0. Penrose, A. Buhagiar, J. Stat. Phys. 30 (1983) 219 [24] J.M. Ball, J. Carr, 0. Penrose, Commun. Math. Phys. 104 (1986) 657 [25] M. Kreer, accepted for publication in Stoch. Process. Appl. [26] F. F. Abraham, Homogeneous nucleation theory - the pretransition theory of vapour condensation, Advances in Theoretical Chemistry, Academic Press, New York-London 1974 (271 D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer, Berlin-Heidelberg 1981 [28] M. Reed, B. Simon, Methods of modern mathematical physics, Vol. I, IV, Academic Press Inc., San Diego, California 1980 [29] T. 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