Metastable states for the Becker-Döring cluster equations

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The Becker-Döring equations, in which cl(t) can represent the concentration of l-particle clusters or droplets in (say) a condensing vapour at time t, are {Mathematical expression} with {Mathematical expression} and either c1=const. ('case A') or {Mathematical expression}=const. ('case B'). The equilibrium solutions are cl=Qlzl, where {Mathematical expression}. The density of the saturated vapour, defined as {Mathematical expression}, where zs is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficients al and bl, there is a class of "metastable" solutions of the equations, with c1-zs small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An "exponentially long time" means one that increases more rapidly than any negative power of the given value of c1-zs (or, in case B, ?-?s) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large. © 1989 Springer-Verlag.

Original languageEnglish
Pages (from-to)515-541
Number of pages27
JournalCommunications in Mathematical Physics
Issue number4
Publication statusPublished - Dec 1989


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