The Becker–Döring Process: Pathwise Convergence and Phase Transition Phenomena

Abstract

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Quasi-stationary distribution and metastability for the stochastic Becker-Döring model

“…for all C ∈ E, where f i is given in (2.5), characterizes the long-time behaviour of any finite-dimensional distributions. Note that ∞ i=2 f i = ∞ for z > z s which suggests that the limiting size distribution has a fat-tail (infinite mean number of clusters above any size), interpreted as a phase-transition, see [5] on a related model. From now on, we…”

“…We refer to our review [4] for historical comments and detailed literature review on theoretical results from the deterministic side. See also [5,10] for recent results on functional law of large number and central limit theorem.…”

“…The model was initially designed to explain critical phase condensation phenomena where macroscopic droplets self-assemble and segregate from an initially supersaturated homogeneous mixture of particles, at a rate that is exponentially small in the excess of particles. This led to important applications in kinetic nucleation theory [9,5]. Mathematical studies in the 90's showed that (in the deterministic context), departing from certain initial conditions, the size distribution of clusters reaches quickly a metastable configuration composed of "small" clusters, and remains arbitrary close to that state for a very large time, before it converges to the true stationary solution that leads to "infinitely large" clusters (interpreted as droplets) [8,9].…”