The physics of boundary conditions in reaction-diffusion problems

TL;DR

This paper introduces a self-consistent method to derive boundary conditions in reaction-diffusion problems, providing deeper physical insight and addressing limitations of standard approaches in modeling chemical reactions at boundaries.

Contribution

The authors present a general approach to derive boundary conditions from reaction schemes and geometry, improving upon traditional methods in reaction-diffusion modeling.

Findings

Recovered known results in specific limits

Provided a physically insightful framework

Demonstrated applicability to paradigmatic examples

Abstract

The use of fully or partially absorbing boundary conditions for diffusion-based problems has become paradigmatic in physical chemistry and biochemistry to describe reactions occurring in solutions or in living media. However, as chemical states may indeed disappear, particles cannot, unless such degradation happens physically and should thus be accounted for explicitly. Here, we introduce a simple, yet general idea that allows one to derive the appropriate boundary conditions self-consistently from the chemical reaction scheme and the geometry of the physical reaction boundaries. As an illustration, we consider two paradigmatic examples, where the known results are recovered by taking specific physical limits. More generally, we demonstrate that our mathematical analysis delivers physical insight that cannot be accessed through standard treatments.