A New Derivation of Robin Boundary Conditions through Homogenization of a Stochastically Switching Boundary

We give a new derivation of Robin boundary conditions and interface jump conditions for the diffusion equation in one dimension. To derive a Robin boundary condition, we consider the diffusion equation with a boundary condition that randomly switches between a Dirichlet and a Neumann condition. We prove that, in the limit of infinitely fast switching rate with the proportion of time spent in the Dirichlet state, denoted by ρ, approaching zero, the mean of the solution satisfies a Robin condition, with conductivity parameter determined by the rate at which ρ approaches zero. We carry out a similar procedure to derive an interface jump condition by considering the diffusion equation with a no flux condition in the interior of the domain that is randomly imposed/removed. Our results also provide the effective deterministic boundary condition for a randomly switching boundary.