# Cluster equations for the Glauber kinetic Ising ferromagnet I. Existence and uniqueness

код для вставкиСкачатьAnn. Physik 2 (1993) 720-737 Annalen der Physik @ Johann Ambrosius Barth 1993 Cluster equations for the Glauber kinetic Ising ferromagnet: I. Existence and uniqueness Markus Kreer * Department of Mathematics, Henot-Watt University, Riccarton, Edinburgh EH 14 4AS, United Kingdom Received 16 July 1993, accepted 31 August 1993 Abstract. The infinite set of cluster equations, proposed by Binder and Miiller-Krumbhaar for a Glauber kinetic Ising ferromagnet in 1974, generalize the Becker-Doring equations used in classical nucleation theory. For positive symmetric transition rates satisfying certain growth conditions and a detailed balance condition we prove for sufficiently fast decaying initial cluster distributions the existence of a positive cluster distribution with finite density for all finite times solving the cluster equations. Uniqueness is proven under some further conditions on the transition rates. Our existence and uniqueness results apply e.g. for a Glauber kinetic Ising ferromagnet in two dimensions. Keywords: Dynamics of first-order phase transitions; Generalized nucleation theories; Nonequilibriurn thermodynamics; Infinite systems of ordinary differential equations. 1 Introduction A thorough understanding of nonequilibrium phenomena in thermodynamics and statistical physics is one of the challenging tasks in modern physics. An example of these nonequilibrium phenomena is metastability, which is usually observed in the dynamics of first-order phase transitions [l]. In the classical approach to metastability a qualitative (and sometimes quantitative) description is provided by nucleation theory dealing mainly with the rate of nucleating clusters [2-61. The Becker-Doring theory, dating back to 1935 [2, 31, is the classical nucleation theory and considers the kinetics of cluster formation. In the dynamical formulation by Frenkel in 1946 [4] the kinetics of clusters is described by a stochastic one-step process-(see, for example [7]) using an evaporation-condensation mechanism, in which a cluster of size j , say, gains or loses a single monomer at a time. For a rigorous mathematical discussion of this stochastic process and the related infinite system of ordinary differential equations we refer to [8] and [9]. Clearly, an improvement of the Becker-Doring theory should allow for general coagulation-fragmentation processes in which two clusters of different size can coagulate or a cluster can split up into two smaller clusters, say. This has been done by Binder and Muller-Krumbhaar in 1974 [lo],who investigated the relaxation of a twodimensional Ising ferromagnet with Glauber kinetics after a sudden reversal of the * Present address: Research & Product Development, Trading Risk, NatWest Markets, 135 Bishopsgate, London EC2M 3UR, U.K. 72 1 M. Kreer, Cluster equations for the Glauber kinetic Ising model, I. applied magnetic field in Monte Carlo studies. In order to explain their findings an approximate theory of cluster dynamics was derived from a master equation, using Fisher’s static cluster model for the desired equilibrium cluster distribution [I I]. Their equations read in the terminology of the coagulation-fragmentation cluster equations discussed for example in [12] as follows where co(t)= 1. The cluster currents W;,k are defined as w;,k= a;,kZC,Ck-b;,kC;+k+l = Wk,; , j , k = 0,I , . . . . (1 -2) Note that here the transition rates are positive constants which are independent of time t and symmetric, that is, a;,k = ak,; and b,,! = b , ; . In this cluster current the first term describes the coagulation of an k- and j-cluster linked together by a newly created I-cluster to make a ( j + k + 1)-cluster, while the second term describes the reversed process of a fragmenting u + k + 1)-cluster. The Coagulation rates a;,k and the fragmentation rates bj,k satisfy a detailed balance condition. The factor in (1.1) is the usual factor to avoid double-counting of clustering-events in the first sum. Unfortunately there does not yet exist a satisfactory microscopic theory for the transition rates a;,k and b,,k (see, also [lo]). However, let us consider as a motivating example a Glauber kinetic Ising ferromagnet on a square lattice in two dimensions. By the Glauber dynamics two clusters of size j and k, say, can coagulate to make a (j+k+ 1)-cluster by bondlinking through a single spin-flip. Therefore the flipped spin needs to connect a surface spin of the j-cluster with a surface-spin of the k-cluster. Hence the coagulation rates a;,k will be proportional to the product of the corresponding cluster surfaces for large j and k. Assuming compact clusters this leads in two dimensions to + a;,$ = O((jk)1’2) as j,k-+co The fragmentation rates b;,k are related to the coagulation rates aj,k by a detailed balance condition. Using the static Fisher-Kuhrt droplet model [I 1, 13, 141 we may write where A > O and O < p < 1. Our hypotheses on the transition rates are motivated by this example and will reflect these properties. The Becker-Doring equations are recovered from (1.1) - (1.2) by taking aj, = bj, 70 for min u,k ) r 1, setting a. = ~ , , ~ / 2 za,, = a,,, ( j= 1,2, . . .) and b;+l= bj,o 0 = 0,1, . . .), and finally shifting the index j to j+1 c, = ( a j - l z c j - l-b,ci)-(ajzc,-bi+lcj+,) , j = 2,3, . . . . (1 -3) In Eq. (1.3) we interpret z as the concentration of monomers (that is clusters of size I), which is kept constant in classical nucleation theory (e.g. [2, 3, 5, 151). ‘ For the finite version, which can be studied numerically on a computer, existence and uniqueness are guaranteed by standard results in the theory of ordinary differential equations (see also Chapter 2). Ann. - Physik 2 (1993) 722 From the mathematical point of view it is a priori not clear whether or not the cluster equations (1 .I) - (1.2) possess solutions We mention in particular that examples of non-existence are known for similar infinite systems of differential equations [12, 16, 171 and that some of these examples are even used as physical models. Therefore it is not only of academic interest to study existence and uniqueness questions. In comparison to the better known coagulation-fragmentationequations (e.g. [121) the equations considered here do not conserve the density p = xjjc,(t). Therefore the sophisticated arguments given in [12], depending crucially on the conservation of density, do not apply in our case to prove existence of a solution for the initial value problem corresponding to the infinite set of ordinary differential equations (1.1)- (1.2). The outline of this paper is as follows. In Chapter 2 we set up the mathematical tools and give some preliminary results. The existence proof is given in Chapter 3 and the uniqueness proof is contained in Chapter 4. Chapter 5 deals with an application of the theorems and contains a final discussion. '. 2 Mathematical preliminaries Following [12] the properties of the solution will be discussed within the framework of the following Banach sequence space a IlYll<ml X=tv=Qr): IlYll = c 4Y,l I= 1 - We write y r O if y r 2 0 for each I = 1,2,. . . and set x+= t v € x : y z o ] . The next definition is motivated by the definition of a solution for the density conserving coagulation-fragmentation equations discussed in the work of Ball and Carr [12]. Definition 2.1 Let O< T I 03. A solution c = (cr)of the cluster equations (1.1)-(1.2) on [0,T ) is a function c: [0,T)+X such that (i) c ( t )2 0 for all t E [0,r) (ii) each cj: [O,r)-+IR is continuous, and s ~ p , , ~ ~ , ~ , \ \ c<( 00 t)ll (iii) f o r all t E [0,T ) ,j = 0, 1, . . . t t m 50 dr k C= O aj,kcj(r)Ck(T)< O0, a j dr C 0 b j , k C j + k + l ( T ) < 03 k=O (iv) for ail t E [0,T), j = 0, 1 , . . . t i (P c cj(t)=Cj(O)+jdr [L y-k,k-l(c(r))W/,k-I(C(r)) 0 2k=1 k= I 1 * It follows easily from the above definition that if c is a solution on [0, 7') then each cj is absolutely continuous, so that c satisfies (1.1)-(1.2) for almost every t E [0,r).In fact, under some stronger assumptions on the initial conditions and the transition rates A numerical study of the cluster equations considered here is subject of a forthcoming paper [18]. M. Kreer, Cluster equations for the Glauber kinetic king model, I. 723 we shall be able to prove that the cj are even continuously differentiable in t. However, for the existence proof we will use the integral version given in (iv) of Definition 2.1. As in earlier work [8, 12, 19,201 on similar equations we prove existence of solutions by taking a limit of solutions of a finite dimensional auxiliary system The next lemma will be an essential ingredient for the existence proof. Lemma 2.2 Let N be any positive integer and K some positive number. For positive symmetric transition rates aj,k and bj,k such that aj,k5K U + k + I) and non-negative initial data ~j"(0) = cj(0)?O , j = 1 , . . .,N the auxiliary system (2.1) - (2.3) hasfor t 2 0 exactly one solution which is non-negative. Moreover, if (gj)ieN is a sequence of real numbers with go = 0, then the aforementioned solution satisfies the identity where the region of summation is R = {tiy k ) :0 5j , k and ti+k ) c N- 1). Prooj The right side of the auxiliary system (2.1) - (2.3) is continuously differentiable in c { ~. .,.,c!(? and therefore it satisfies a local Lipshitz condition. Hence by standard theorems (e.g. [21]) there exists locally a solution to the initial value problem which is unique. Recall that this local solution exists or times t E [0,T ) ,(where T> 0 is the maximum existence time), is continuously differentiable in t, satisfies the auxiliary system (2.1)-(2.3) and takes the initial values at t = 0. That this solution is non-negative follows by an argument given for example in [21], see also [20]: Consider for E > O the solution c{";"), . . . of the system obtained by adding E to the right-hand sides of the Eqs. (2.1) and satisfying the same initial condi) t and some k e { l , .. .,N) such that tions. By considering the sign of c i N E ) ( tfor c l ! " ( t ) = 0, cjN;"'(~)?O,for 0 s r l t and j # k, it follows easily using standard results on ordinary differential equations that cj!"'(t) is non-negative and tends as e+O+ to a non-negative solution cp(t)of (2.1)- (2.3) defined for all t in some interval [ O n , T> 0. Before showing global existence (i.e. for the existence time T = +m) we shall derive the identity (2.4). From the auxiliary system (2.1)-(2.3) we deduce that ,cp"' 724 Ann. Physik 2 (1993) After some algebra the first term on the right side in (2.5) can be written as where R = {(j,k) :0 ~ jk and , (j+k)IN- I ] . Further we can use the symmetry in j and k to write the second term on the right side in (2.5) as For the first term on the right-hand side of Eq. (2.7) we obtain after some algebraic manipulations, using go = 0 and cj.w(t) 0 for j > N , For the second term on the right-hand side of Eq. (2.7) we obtain in the same way, interchanging the rdles of j and k, The desired identity (2.4) is then obtained from (2.5) by combining (2.6), (2.7), (2.8) and (2.9). Global existence of the solution can now be deduced from the identity (2.4) as follows: taking g . = j + 1 if j 2 1 and go = 0 we derive from (2.4), using the positivity of the solution c : ~ ’ , j = 1 , . . .,N, the positivity of the transition rates u j , k , bj,kand the cut-off condition c j N ) ( t = ) 0 for j > N , 1 c 0’+1)Cjw(t)s-ao,OZ+ c uj,ozCj.N)(t) . dtj= 2 N d N - (2.10) j=1 1 Integrating this last inequality (2.10) with respect to t and using the monotonicity of the Riemann integral leads to j= I j = I 0 j= 1 Thus we conclude from (2.11) after application of Gronwall’s inequality, using Uj,0I-KO’+ l), eKzt (2.12) M. Kreer, Cluster equations for the Glauber kinetic king model, I. 725 where the right hand in (2.12) defines a positive and continuous function in t. Thus each cjN)(-) (j = 1,2, . . .,N)is bounded below and above by continuous functions and cannot blow up in finite time. Global existence follows by the standard result that either the solution exists globaly (that is, for all times t 10) or blows up in finite time (see, Q.E.D. for example [21]). This concludes the proof. The next theorem is a consequence of the previous lemma. Theorem 2.3 Let A, K, A be positive numbers. Let the transition rates satisfy O<a;,,<KU+k+l) andO<b;,,<K(j+k+l) forj,kiO, andlet theinitial values to the auxiliary system (2.1) - (2.3) satisfy Oscf”(O)<A(l+A)-’ , EN . Then there exist positive continuous functions A p ( t ) ,p N such that for t r O E N, which are independent of (2.13) where the cjN)is solution of (2.1)- (2.3) to the prescribed initial values. Pro08 The proof will be done by induction via p. From Lemma 2.2 we know, that there exists a unique, non-negative, continuously differentiable solution of the finite auxiliary system (2.1) - (2.3), satisfying the initial values and the identity (2.4). For the first induction step we consider p = 1. We derive from the identity (2.4) in Lemma 2.2, in analogy to the proof of this lemma, using the positivity of the solution cjNN’, j = 1, . . .,N, the positivity of the transition rates a,,k, bj,, and cjNN,(t)=Ofor j>N, (2.14) where A, (.) is a continuous, positive function in t, independent of N,defined as We set A$@):=l l ( t ) so that also c CJ”(t)l&(f) N 01 ;= (2.15) 1 and the first induction step is finished. Next we show, that if the bound (2.13) holds for 1 , . . . , p - I it also holds for p , Ann. Phvsik 2 (1993) 726 c j % j N ) ( t ) r d p ( t ). N O r (2.16) j = 1 Therefore assume that the inequalities (2.13) are already proven for 1, . . .,p-1. Then we conclude from (2.4) with gj = j p if j ? 1 , using the positivity of the solution cjN), j = 1, . . .,N, the positivity of the transition rates aj,k, bj,kand cjN)(t)=0 for j > N , since a,,krKG+k+l). From this last line we obtain after some algebraic computations and rearranging the terms, using the induction assumption and the binomial theorem, (2.17) where the positive function 9pp-l(f) is defined as Applying Gronwall’s inequality on the integrated version of (2.17) and using the decay condition on the initial data proves the bound (2.16), taking and concludes the induction. Q.E.D. 3 Existence Before stating our existence theorem we shall give a useful proposition. Proposition 3.1 Under the conditions of Theorem 2.3 the function family of solutions to the auxiliary system (2.1)-(2.3), (Cj”)(f))jeR.l,N€R.l, is equicontinuous on [o,TI for any T>O. Pro03 From the bound on the cIN) given in (2.13), Theorem 2.3, we derive for j = 1,2,. . . 727 M. Kreer, Cluster equations for the Glauber kinetic Ising model, I. . O I C j ” ’ ( t )-42 1 9( t- ) , t>O , p € N jp (3.1) Since (j+l ) / j r 2 for j ? 1 , we find for j = 0,1,. . . where dp(t) = 2p-42p(t)+1. Utilizing this bound (3.2) in the auxiliary system (2.1)- (2.3) we find for j = 1, . . .,Nand t E [0,TI (T>0 arbitrary but fixed), using the hypotheses on the transition rates, after some lengthy computations ( sKz[4(t)12 For p = 3,4, i+l)2+K4(f) k = 1 kP-’ ( k=t - +1l ) k”-’ . (3.3) . . . we can define continuous functions so that with (3.4) the inequality (3.3) can be written as :1 I -cjN)(t) r Q p ( t ) , p = 3,4, . . . . (3.5) Since the derivatives are uniformly bounded by (3.5) for p 2 3 and for all j E N, N E N on [O,T],T>O,we may integrate the inequality (3.5) with respect to t from t , to t2, where t,, tz E [0,TI, and use the monotonicity of the Riemann integral r ( sup Qp(t))It2-tlI , j , N E N , p = 3 , 4 ,... b,.‘21 The last chain of inequalities implies the desired equicontinuity on [0, TI. . (3.6) Q.E.D. Now we can state our existence theorem. Theorem 3.2 Let there be a K>O such that O<aj,,CK(j+k+1) and O<bj,k<K(j+k+l)f o r j , k i O . Let there bepositive numbersA and A and let the initial cluster distribution satisfy Orcj(0)<A(l+A)-’ , j E N . Then the cluster equations (1.1)- (1.2) admit at least one non-negative global solution which is continuously differentiable, satisfies the initial conditions and has the finite moment property for all finite times, that is for any T>O 728 Ann. Physik 2 (1993) Pro08 We consider the integrated version of (2.1)-(2.3) which reads as t 0 N-j+ I j Cjw(t)=Cj(0)+Ed? C [:k= $!!lpk-11 C k= 1 1 Wj.$)-l j=I,...,N (3.8) where WJ$) is defined in (2.3), cjM = 0 for j > N , and the initial values are chosen to satisfy cj”(0) = cj(0).Then we show that one can approximate with this system the infinite version (obtained by integrating (1 .I)) (3.9) uniformly on [0, T ] for any T>O. And we show also that for a suitable sequence of positive integers N the c”: converge uniformly on bounded intervals of the positive taxis towards non-negative continuous functions cj. These are then the desired solutions to the infinite system (3.9) with kernes defined in (1.2)To do this recall that the solutions of (2.1)-(2.3) with the initial values ~(”(0) = cj(0) (j= 1, . . .,N) solve also the system of integral equations (3.8) with kernel (2.3) and by Proposition 3.1 constitute an equicontinuous function family on [0,TI, T>O. We shall now apply Arzela-Ascoli’s theorem together with the “diagonal” argument described by McLeod “191, see also [8, 201) to show that there exists a sub-sequence converging uniformly to a solution of the infinite system (3.9). Let T be any positive number. Arzela-Ascoli’s theorem shows that there is a sequence of integers S1 C N such that the sequence of functions clN”(t) (Nl E S,) converges uniformly on the interval of the positive t-axis to a continuous function c, ( t ) . A second application of Arzela-Ascoli’s theorem shows that there is a further subsequence S, C S,, such that ciN2)(t) (Nl E S ~ )converges uniformly to a continuous function c 2 ( t ) ,and so on. Thus we can construct by this “diagonal” argument a chain of nonempty sets satisfying S j + l C Sj. In the limit j - 03 we obtain a nonempty set S, C N such that we can define a sequence of continuous functions cj(t),j = 1,2, .. ., by cj.”(t)+cj(t) as N+m , N E S , (3.10) where the convergence is uniform on [0, TI for any T>O. It is important to note that also the limiting functions in (3.10) satisfy the inequalities from Theorem 2.3 for j = 1,2, . . . (3.11) and furthermore there exist continuous functions Mp( t ) such that (3.12) M. Kreer, Cluster equations for the Glauber kinetic Ising model, I. 729 c (take for example M,(t) = Ap+2(f) j j -’). This proves the finite moment property as stated in the theorem. With co(f)= 1 and using ( j + l ) / j s 2 for j z 1 we obtain for j = 0,1, ... (3.13) Now we pass to the limit in the auxiliary system of integral equations (3.8) with kernels (2.3), restricting N to values in the set S, C N. Let & > O be given and consider the time interval [0,TI for arbitrary T > 0. Then for any j E N we can choose some positive integer N,(&, T ) E s, such that for all N > N , (&, T ) with N E S, because the terms in (3.14) are finite sums and the uniform convergence property as stated in (3.10) applies directly on [0,TI. Next we have to show that also for arbitrary j E N there exists some positive integer N2(e,T ) such that the following estimate holds (3.15) To see this consider the following argument. For the coagulation terms we find at once the upper bound (3.16) For E E 0 and T > 0 given we can fix a positive integer M’ = MI(&,T ) such that, using our bounds (3.2) and (3.13) on the cJNN’(t) and c j ( t ) respectively, (3.17) wherep = 3,4, . . . and where we also used the hypotheses on the coagulation rates. Now for the first term in the inequality (3.16) we can choose some positive integer N ; ( E ,T ) such that for all N>N2(&,T),N E S , by the Arzela-Ascoli theorem and the “diagonal” argument, (3.10), 730 Ann. Physik 2 (1993) (3.18) Combining (3.17) and (3.18) implies that we can estimate the coagulation terms in (3.16) T), N E S , for E > O and T>O given, such that for all N>N2(&, (3.19) In the same way we obtain for the fragmentation terms for E > 0 and T> 0 given, that there exists a positive integer N’(E,T ) such that for all N>N;’(&,T), N E S , (3.20) Setting N2 = max [N$,W;)we see that for E>O and T>O given for all N > N 2 , N E S, the desired estimate (3.15) holds. From (3.14) and (3.15) we see that the convergence is uniform for t E [0, TI for N>max [N,,N2Jand N E S , , t E 10, TI. The estimate (3.21) shows that the right sides of the infinite system of integral equations (3.9) can be approximated uniformly on [0, T ] by the right sides of the finite auxiliary system (3.8). That the left sides of the infinite system of integral equations (3.9) can be approximated uniformly on [0, TI by the left sides of the finite auxiliary system (3.8) follows easily by (3.10). Thus we have shown that the infinite system of integral equations (3.9) with the kernels defined in (1.2) has a solution on [0, T ] for T>O. Since the convergence is uniform and the limiting functions obtained in (3.10) are continuous, the integrands in (3.9) are also continuous. Thus differentiation with respect to t shows that c j ( * ) is a continuously differentiable solution of the infinite - (1.2) on [0, TI, satisfying the initial conditions. system of differential equations (1 .l) Q.E.D. Remark: For the proof of existence of a continuously differentiable solution to (1.1)-(1.2) it is sufficient to require that the initial conditions satisfy even a weaker decay condition, namely j”= J P cj(0) < 03 for some positive integer p 2 3. c Corollary 3.3 Let the conditions of Theorem 3.2 be satisfied. Then for any positive integer N and f o r any real sequence (g,)jaNwith go = 0 the solution of the cluster equations (1.1) - (1.2) satisfies the identity c d N 1 - g;c; = (gj+k+t-gj-gk)y,kdtj=i 2 Os;+ksN-l c N a c g; c j = l k=N-j y , k - (3.22) M. Kreer, Cluster equations for the Glauber kinetic Ising model, I. 73 1 If there is any positive integer n such that Igi/jn 1 is bounded for all j r 1, then in the limit N+ 00 (3.22) simplifies to the following identity Pro03 The first identity (3.22) is derived by the same computations as in the proof of Lemma 2.2. The second equation (3.23) is derived from Eq. (3.22). By the finite moment property, from Theorem 3.2 and the Weierstrass uniform convergence argument one can show after some trivial but tedious computations that uniformly on [0, TI for some n E iN. Q.E.D. Remark: Let us consider the special case of transition rates aj,k= 1 and bj,k'O for j , k = 0,1, . . . . Taking z>O and for example monodisperse initial conditions, that is, c1(O)>O and c,(O) = 0 for j = 2,3, . . . we find from Corollary 3.3, Eq. (3.23), with g j = j + l f o r j > l and g,=O U - c O'+i)c;(t) = z + c c ; ( t ) l Z dtj=1 ;= 1 and thus This example demonstrates explicitly that the density p is not a conserved quantity for the cluster equations (1 .l)-(1.2) as compared with the situation in the coagulationfragmentation equations considered in [121. 4 Uniqueness This section contains a uniqueness result which includes the physically interesting cases. The uniqueness theorem and its proof are analogous to the one given in [12]. Theorem 4.1 Let K > 0 and 0 5 a I1 /2 and assume that (i) O<aj,,<K ((j+l)(k+l))" (ii) c h,=OJ ( r ) .1-ab r - j - l , , s K r * - a ,r z l where h ( r ) = int [(r+1)/2] denotes the integer part of (r+ 1)/2. 732 Ann. Physik 2 (1993) Let c,(O) satisfy the decay condition of Theorem 3.2 and let T>O. Then there exists exactly one solution c: I?+ -tX+of (1.1) - (1.2) on [0,T ) satiyfying the initial conditions. Pro03 Global existence is guaranteed by the previous existence theorem since (i) ((j+1)(k+l))a<2(j+k+1) for Osai+ (ii) b,-,-l,j<Kr*-a V r E N, 1 r j l r , implies b k , j < K ( j + k + l ) ,where k = f - j - 1 . Now let c and d be the two solutions of (1.1)-(1.2) on [0, T ) satisfying cj(0) = dj(0) for j = 1 , 2 , . . . . Set x = c-d and let p = 1-a, and We show that there is a positive, continuous function K ( - )such that so that, by Gronwall’s lemma, d ( t ) = 0, and c = d. From Corollary 3.3, Eq. (3.22), we obtain (4.3) where, using xo = 0, N m We first estimate UN.Notice that we have M. Kreer, Cluster equations for the Glauber kinetic king model, I. I 4KZ (1 + 4M1 (f )) 19 (L‘ ) 733 (4.8) where we have used a I1 - a = 6, the upper bound on c j j c j , (3.12), and the definition of 19(.) in (4.1). The same estimate is obtained for the terms involving dkxj, since the bound (3.12) holds equally for c and d, so that The inequalities (4.8) and (4.9) are the desired estimates for the coagulation terms in U N . To estimate the fragmentation terms in U N , note that -(O’+k+l)Bsgn (X,+k+l)-jBsgn(xj)-kBs g n ( X k ) ) X j + k + l s0’8+ka-O’+k+1)B)lxj+k+l I and therefore we find (4.10) Also with h ( r ) = int [(r+1)/2], = 1 - a and the with ar-j-,j = (r-j-l)B+j’-r8sj8. hypothesis (ii) on the fragmentation rates bj,k we conclude from r- 1 C ;=0 c a,_j-l,jbr-j-l,jc2Kri-a h(r) ar-j-1,jbr-j-1+js2 j=O that the following inequality holds (4.11) 734 Ann. Physik 2 (1 993) Combining all estimates, namely (4.8), (4.9) and (4.11) we obtain from (4.4) for all N E N that (4.12) where = 8Kz(1+4M1(r))+2K . K(?) (4.13) Next we argue that t 1dr V , ( t )= 0 N-tm 0 lim (4.14) and the desired inequality (4.2) follows readily from (4.3) using (4.12) and (4.14). But (4.14) follows easily from (4.5) by the bound (3.13) and Weierstrass' uniform convergence argument: First estimate the coagulation terms in V, because the following sum converges for p > 3 = "c "c -1 N = 1j = ~ J 1 P (- ~~ - j + 1 ) ~ - ~ cy=lj-@-2)(N-j+1)-@-2)must tend to 0 as N-tm. and thus uniform on The convergence is [o, 7"). Exactly the same argument is used for terms involving dkxjto obtain c jasgn (xj> c m ;= 1 a,,kd,*,l-+o as N+m , p = 4 , 5 , . . . (4.16) k=N-j where the convergence is also uniform on [0,7'). Finally estimate the fragmentation terms in V, using hypothesis (ii) of the theorem and the absolute convergence of the infinite series by (3.13) M. Kreer, Cluster equations for the Glauber kinetic Ising model, I. 735 if p>2. The convergence here is uniform on [0, r). The inequalities (4.15), (4.16) and (4.17) prove the desired limit (4.14). This concludes the proof. Q.E.D. 5 Concluding remarks Combining the results in the previous section we can prove that the cluster equations of Binder and Muller-Krumbhaar have a global, unique, non-negative solution for the class of transition rates suggested by physical considerations for an Ising ferromagnet in two dimensions. For coagulation rates proportional to the cluster surface and fragmentation rates satisfying a detailed balance condition, where the underlying equilibrium cluster theory is the Fisher droplet model, our existence and uniqueness theorems hold. We state this fact as a Corollary 5.1 Let the transition rates be such that for j , k r 0 there are positive constants A. and Bo such that bj, k = BO aj, k [(j+k+1 ) P - j P -kP J o’+ k+ ’ o’+ 1)5(k+1) ’ for some constants 0 <p < 1, 5 >0. And let there be positive constants A and A such that the initial cluster distribution satisfiies Oicj(0)<A(l+A)-i , j e N . Then the cluster equations (1.1)- (I.2) have exactly one non-negative global solution which is continuously differentiable, satisfies the initial conditions and has the finite moment property for all finite times, that is, for t r O c jPCj(t)<rn , p € N . Ca 01 j = 1 Pro08 To apply Theorem 4.1, which guarantees existence and uniqueness, we need to check the conditions stated there. Thus we only need to verify hat the transition rates aj,k and bj,k satisfy the hypotheses required there. Obviously the coagulation rates aj,k satisfy the growth condition (i) of Theorem 4.1. To do it for the fragmentation rates bj,kchoose an integer ro>l and fix a positive number B such that O< B < 1+ (-y - (1 +-) P . Hence for r z ro and j = 1,2, . . .,h ( r )I(r+ 1)/2 (where h ( r ) = int [(r+ 1)/2])we find that 736 Ann. Physik 2 (1993) This leads us to rp-jp-(r-j-l)ps-Bjp+l , j = 1 , . ..,h(r) , r z r o . (5.1) Furthermore we have u+k+ 1)7 51. (j+l)‘(k+ l ) T Applying the inequalities (5.1) and (5.2) to the given fragmentation rates bj,k we see that the following inequalities hold This shows that the condition (ii) on the fragmentation rates bj,kfrom Theorem 4.1 are satisfied and application of Theorem 4.1 concludes the proof. Q.E.D. The phenomenological cluster equations considered in this paper for the description of the dynamics of an Ising ferromagnet with Glauber kinetics in two dimensions, have been very helpful in understanding results of Monte Carlo studies (e.g. [ l o ] , see also [6]).Since a derivation of the stochastic Glauber dynamics for Ising models from a quantum mechanical model is possible (e.g. [22]) one might be able to establish equations such as (1.1)-(1.2) for a Glauber kinetic Ising ferromagnet in certain regimes of the phase diagram from a more rigorous point of view. We mention again that the Eqs. ( 1 . l ) - (1.2) represent a generalisation of the linear Becker-Doring cluster equations (1.3). These equations and the related classical theory of nucleation proved very useful in physics and chemistry and hence the question arises whether or not these generalised cluster equations exhibit the same phenomena as for example metastability and nucleation, algebraic cluster growth at late stages, etc. It could perhaps happen that the possible variety of clustering processes destroys the slow time evolution as observed in the one-step process in [8] and [9].We recall the similar problem whether or not an infinite Ising ferromagnet with Glauber dynamics can support metastable states, [ l , 23, 24, 251. An other open problem is related to the problem of gelation in the “density-conserving” coagulation-fragmentation equations (e.g. [ 12, 261). As there our global existence proof fails if the coagulation rates grow too fast, e.g. as aj,k j ” k a where 1/2 < a < 1 There the solution loses its analyticity in time t at a certain time T, (e.g. [19, 261) and this phenomenon is associated with a phase-transition: gelation. - ’. Note that a Glauber kinetic Ising ferromagnet in three dimensions would suggest the choice a = 2/3. It is well known that the dynamic properties of Ising ferromagnets in two and three dimensions are qualitatively quite different [271. M. Kreer, Cluster equations for the Glauber kinetic king model, I. 737 With our methods we can still prove local existence, that is, existence up to a time To>O, for aj,k - j u k u where 1/2 < a < 1 and bj,k related to aj,kby a detailed balance condition, using the Fisher-Kuhrt droplet model. Unfortunately the global result cannot be achieved because certain moments blow up. At present we do not know whether or not even the first moment, that is, the density p = z j j c , ( t ) blows up in finite time but we shall not enter in a discussion of this problem here. References [I] 0. Penrose, J. L. Lebowitz, Towards a rigorous molecular theory of metastability, in: Studies in Statistical Mechanics VII (“Fluctuation Phenomena”), E. W. Montroll, J. L. Lebowitz (eds.), North-Holland, Amsterdam 1976. 1987 [2] R. Becker, W. Doring, Ann. Physik 24 (1935) 719 [3] R. Becker, Theorie der Warme, Heidelberger Taschenbucher Band 10, Springer-Verlag, Berlin Heidelberg 1978 [4] J. Frenkel, Kinetic theory of liquids, Oxford University Press, London 1946 [5] F. F. Abraham, Homogeneous nucleation theory - the pretransition theory of vapour condensation. Advances in Theoretical Chemistry, Academic Press, New York- London 1974 [6] 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, New York-London 1983 [7] N. G. van!Kampen, Stochastic processes in physics and chemistry, North-Holland, Amsterdam 1981 181 0. Penrose, Commun. Math. Phys. 124 (1989) 515 (91 M. Kreer, Ann. Physik 2 (1993) 398 [lo] K. Binder, H. Miiller-Krumbhaar, Phys. Rev.-B9 (1974) 2328 [ l l ] M.E. Fisher, Physics 3 (1967) 255 [12] J.M. Ball, J. Carr, J. Stat. Phys. 61 (1990) 203 [I31 F. Kuhrt, Z. Physik 131 (1951) 185 [14] F. Kuhrt, Z. Physik 131 (1951) 205 [151 J. Feder, K.C. Russel, J. Lothe, G.M. Pound, Advan. Phys. 15 (1966) 1 1 1 [161. J. Carr, F. P. da Costa, ZAMP 43 (1 992) 974 [17] F. P. da Costa, PhD-Thesis, Heriot-Watt University, Edinburgh 1993 [IS] M. Kreer, in preparation [19] J.B. McLeod, Q. J. Math. 13 (1962) 119 [20] J.M. Ball, J. Carr, 0. Penrose, Commun. Math. Phys. 104 (1986) 657 [21] P. Hartman, Ordinary Differential Equations, Wiley, Baltimore 1973 [22] Ph.A. Martin, J. Stat. Phys. 16 (1977) 149 [23] D. Capocaccia, M. Cassandro, E. Olivieri, Commun. Math. Phys. 39 (1974) 185 I241 G.L. Sewell. Quantum Theory of Collective Phenomena, Clarendon Press, Oxford 1986 [25] R. Schonman, Reprint 1261 R.L. Drake, A general mathematical survey of the coagulation equation, in: Topics in Current Aerosol Research, Vol. 3, Part 2, G.M. Hidy, J.R. Brock (eds.), Pergamon Press, Oxford 1972, p. 201 - 376 [27] E. Olivieri, private communication

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