**Annalen der Physik**

# Classical Becker-Döring cluster equations: Rigorous results on metastability and long-time behaviour

## Abstract

We consider the classical Becker-Döring cluster equations with constant monomer concentration *c*_{1} = *z* > 0 and

as a model which describes the kinetics of a first-order phase transition. For a large class of positive coefficients *a*_{l} and *b*_{l} (including the ones commonly used in physics and chemistry) we prove the following: (i) When the monomer concentration *z* is slightly greater than *z*_{s} = lim_{l→∞}*b*_{l}/*a*_{l} then all initial states - containing only subcritical clusters of size *l* < *l** (where *l** denotes the critical size of a nucleus and depends on the supersaturation *z*-*z*_{s} > 0) - 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 *T*_{M} ∼ exp (*C*(*z*-*z*_{s})^{−ω}) (where *C* and ω are some positive constants).

(ii) For times greater than the lifetime *T*_{M} this metastable state breaks down in the following sense: as *t*→∞ each of the *c*_{l}(*t*) converges towards the Becker-Döring steady-state solution *f*_{l}(*z*) like *c*_{l}(*t*)-*f*_{l}(*z*) = O (exp (- | λ_{1} | *t*)) (where λ_{1} < 0 is the eigenvalue closest to 0 of a certain infinite transition matrix) and the total mass of supercritical clusters (i.e. clusters of size *l* > *l**) diverges in this limit. For large times the cluster-number increases linearly in time in the sense that lim_{t→∞}*n* (*t*)/*t* = *J*(*z*), where *J*(*z*) > 0 is the Becker-Döring steady-state current. For the average cluster size *l* = Σ*lc*_{l}(*t*)/Σ_{t=1}*c*(*t*), we find for sufficiently large times algebraic growth in time *t*, that is, μ_{1}*t*^{1/(1-α)} < *l*(*t*) < μ_{2}*t*^{1(1-α)} (where 0 < α < 1 is the algebraic growth exponent of the *a*_{l} ∼ *l*^{α} and μ_{1}, μ_{2} are suitable positive constants). This bound covers previous suggestions due to computer simulations and heuristic calculations.