  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.

Classical Analysis and ODEs;Analysis of PDEs