Metastable states for the Becker-Döring cluster equations

The Becker-Döring equations, in which c_l(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 c_l=Q_lz^l, where {Mathematical expression}. The density of the saturated vapour, defined as {Mathematical expression}, where z_s 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 a_l and b_l, there is a class of "metastable" solutions of the equations, with c₁-z_s 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-z_s (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.