Diffusion laws select boundary conditions

TL;DR

This paper demonstrates how diffusion laws can inherently determine boundary conditions in diffusion equations by analyzing the limit of a small diffusivity outside the domain, clarifying their physical meaning.

Contribution

It introduces a method to derive boundary conditions from diffusion laws by extending the domain with a small diffusivity and analyzing the limit, linking macroscopic boundary conditions to microscopic models.

Findings

Diffusion laws can select boundary conditions as diffusivity approaches zero.

The boundary condition type (Dirichlet or Neumann) depends on the diffusion law.

Microscopic random walk models explain the boundary condition selection.

Abstract

The choice of boundary condition makes an essential difference in the solution structure of diffusion equations. The Dirichlet and Neumann boundary conditions and their combination have been the most used, but their legitimacy has been disputed. We show that the diffusion laws may select boundary conditions by themselves, and through this, we clarify the meaning of boundary conditions. To do that we extend the domain with a boundary into the whole space by giving a small diffusivity $\eps>0$ outside the domain. Then, we show that the boundary condition turns out to be Neumann or Dirichlet as $\eps\to0$ depending on the choice of a heterogeneous diffusion law. These boundary conditions are interpreted in terms of a microscopic-scale random walk model.