Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics

TL;DR

This paper demonstrates how reactive boundary conditions in molecular reaction models can be derived as limits of steep interaction potentials, using asymptotic analysis for both Brownian and Langevin dynamics.

Contribution

It provides a rigorous derivation of reactive boundary conditions from interaction potentials, highlighting the importance of scaling in different dynamical models.

Findings

Reactive boundary conditions arise as limits of steep potential barriers.

Different scalings are needed for Brownian and Langevin models to achieve consistent boundary conditions.

The method of matched asymptotic expansions is used to derive the limits.

Abstract

A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial absorption condition, which uses a Robin boundary condition in the separation coordinate between two possible reactants. This boundary condition can be interpreted as an idealization of a reactive interaction potential model, in which a potential barrier must be surmounted before reactions can occur. In this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation, where molecules react instantly upon reaching the peak of the barrier. The limiting boundary condition is derived by the method of matched asymptotic expansions, and shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased. Limiting boundary conditions for the same interaction potential in both the overdamped Fokker-Planck equation (Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are investigated. It is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit where the Kramers equation solution converges to the solution of the Fokker-Planck equation.