The Becker-D\"oring process: pathwise convergence and phase transition phenomena

TL;DR

This paper proves the convergence of the stochastic Becker-D"oring process to its deterministic equations and demonstrates the presence of phase transition phenomena in the stochastic model, supported by numerical simulations.

Contribution

It establishes pathwise convergence of the stochastic process to the deterministic model and reveals phase transition phenomena in the finite stochastic setting.

Findings

Pathwise convergence towards deterministic equations

Presence of phase transition in stochastic model

Numerical illustrations supporting theoretical results

Abstract

In this note, we study an infinite reaction network called the stochastic Becker-D\"oring process, a sub-class of the general coagulation-fragmentation models. We prove pathwise convergence of the process towards the deterministic Becker-D\"oring equations which improves classical tightness-based results. Also, we show by studying the asymptotic behavior of the stationary distribution, that the phase transition property of the deterministic model is also present in the finite stochastic model. Such results might be interpreted closed to the so-called gelling phenomena in coagulation models. We end with few numerical illustrations that support our results.