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

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Ann. Physik 2 (1993) 398-417
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
(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
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
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
for 1 = 2 , 3,...
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)
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
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
o ( fI-)
bl+l b
(iii) -s--[
bl = Z,>O
lim I-+-
(iv) 3G,G’,y,y’>O
with O < y < l :
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, . . .:
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
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
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
with the steady-state current
Ann. Physik 2 (1993)
( 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 '
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)<
a(0) =
/= 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 , . . .
c 12cl(t)sa(0)exp ( 4 A A z t ) c
O s a ( t )=
, tz0
I= 1
With the definition of I* as in (2.4) we have for the following quantities
the bounds
(0)+ to J * + to J*
M. Kreer, Classical Becker-Doring cluster equations
to = Az
fl=- 2 - a
J* = Aai.QI.zP + l
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
( i i ) For A 2 1 fixed, J* is exponentially small as & -' Of,
ing bound holds
i.e. for E E ( 0 , ~ ~the
) follow-
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,. . .
for 1=1,2, ...
- 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
if= ? f , - l X f - , - 61Xf+Y/[Xf+1, I = 2,3, . . .
with the transformed initial data xf(0)= d i c ~ +(0),
l I = 1, . . . and
Ann. Physik 2 (1993)
Using a more compact notation, (2.22)-(2.23) can be written as
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
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
and the initial value problem (2.26) is solved uniquely for x(0)E H by
X(t) = S(t)(x(O)-Xeq)+Xeq
M. Kreer, Classical Becker-Donng cluster equations
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
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
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
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
Ann. Physik 2 (1993)
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
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-
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
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.
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'
... .
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:
Ann. Physik 2 (1993)
( i ) There exist positive constants E ~ KR,
Cg, xg such that for
-(l'y)-- v'y' the following bound holds
E E(0,~~
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
E(t) = (CZ(t),. . .,C" ( t ) )
Then we can write the Becker-Doring cluster equations as
?(t) = A E ( t ) + 5 + i i ( t )
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-
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)=
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)
Now notice that by Lemma VI in the Appendix A is symmetric and thus
because all eigenvalues In,n = 1,2, .. . I*- 1, satisfy
Together with the bound (2.18) on I* we find for E E [O,E,]
Further using (3.2) and Lemma II we obtain
and in the same way
With these bounds (3.13), (3.16) and (3.17) we derive from (3.12) the following estimate
We choose now some positive number KR and define for
E E (O,E,,]
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
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
G'lY. Define for E
E (0, &o)
j-& - ( I - Y ) / Y
E E(0,~~
find from the bound on MI( t ) given in (2.14) for t i TM=
where the positive constants are defined as
and the proof is finished.
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
And in the limit t+a,
lim n ( t )= J ( z )
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
Ann. Physik 2 (1993)
0 5 q ( t )Ia(0)
exp (4AAzt)
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
n ( t ) = n(O)+Sd7(J(Z)+bz[fi(z)-cz(r)l)
With (4.1) we derive the following inequality
and from this
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
I= 1
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
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
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 ) =
If,(z)+M,(t) and (4.2). This concludes
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]):
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
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]
P3= const, - c c o n ~ t ~ ~ - " ~
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 ~ . ]
( 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
Ann. Physik 2 (1993)
has the following bound on its real eigenvalues A,,, n = 1, . . .,N1
with the definition bl =
Under the hypothesis
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
- - - -'
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.
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