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 $\rho$, approaching zero, the mean of the solution satisfies a Robin condition, with conductivity parameter determined by the rate at which $\rho$ 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.