A mathematical justification of the finite time approximation of Becker-D\"oring equations by a Fokker-Planck dynamics

TL;DR

This paper rigorously establishes the finite-time approximation of Becker-D"oring equations by a Fokker-Planck PDE, providing mathematical justification for their relationship in coagulation-fragmentation processes.

Contribution

It proves the connection between Becker-D"oring and Fokker-Planck equations for finite times, based on decay estimates and Taylor expansion control.

Findings

Validated the approximation through decay estimates and numerical simulations.

Established the mathematical link for finite-time evolutions, not just asymptotic limits.

Abstract

The Becker-D\"oring equations are an infinite dimensional system of ordinary differntial equations describing coagulation/fragmentation processes of species of integer sizes. Formal Taylor expansions motivate that its solution should be well described by a partial differential equation for large sizes, of advection-diffusion type, called Fokker-Planck equation. We rigorously prove the link between these two descriptions for evolutions on finite times rather than in some hydrodynamic limit, motivated by the results of numerical simulations and the construction of dedicated algorithms based on splitting strategies. In fact, the Becker-D\"oring equations and the Fokker-Planck equation are related through some pure diffusion with unbounded diffusion coefficient. The crucial point in the analysis is to obtain decay estimates for the solution of this pure diffusion and its derivates to control remainders in the Taylor expansions. The small parameter in this analysis is the inverse of the minimal size of the species.