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

We consider the classical Becker‐Döring cluster equations with constant monomer concentration c 1 = z > 0 andas 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     l ∞ =1lc l ( t )/Σ   l ∞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.