Semi-permeable Interfaces and the Target Problem

Diffusion through semi-permeable interfaces arises in a wide range of processes, including reverse osmosis for desalination, molecular transport in biological cells, porous media, and animal migration in heterogeneous landscapes. In this chapter, we review recent work on first passage time (FPT) problems for a particle to be absorbed within some target domain \(\mathcal {U}\) whose boundary \(\partial \mathcal {U}\) acts as a semi-permeable interface. We begin by considering an interface with a constant permeability \(\kappa _0\) and a target with a constant rate of absorption \(\gamma \) . Quantities of interest such as the survival probability, absorption flux, and the FPT density can be determined by solving a Robin boundary value problem for the forward diffusion equation in Laplace space. We present a general method of solution based on Green’s functions and the spectral decomposition of certain Dirichlet-to-Neumann (D-to-N) operators. We then extend the theory by considering modifications in either the absorption process or the transport process across the interface, both of which are non-Markovian. First, we introduce an encounter-based model of partial absorption within the target \(\mathcal {U}\) . Encounter-based models consider the joint probability density or generalized propagator for particle position and the amount of particle-target contact time prior to absorption. Absorption occurs when the contact time exceeds a random threshold. If the probability distribution of the latter is an exponential function, then one recovers the Markovian example of absorption at a constant rate, whereas a non-exponential distribution signifies non-Markovian absorption. In the case of a partially absorbing target \(\mathcal {U}\) (surface \(\partial \mathcal {U}\) ) the contact time is given by a Brownian functional known as the occupation time (boundary local time). Second, we describe a more general probabilistic model of single-particle diffusion through a semi-permeable interface, by combining the encounter-based approach with so-called snapping out Brownian motion (BM). Snapping out BM sews together successive rounds of partially absorbing BMs that are restricted to either the interior or the exterior of \(\partial \mathcal {U}\) . The rule for terminating each round is implemented using the encounter-based model. We show that this results in a time-dependent permeability that can be heavy-tailed.