Boundary conditions at a thin membrane that generate non--Markovian normal diffusion

Tadeusz Kosztołowicz

Published 2019-10-06Version 1

We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear relation between the probabilities of finding a particle on both membrane surfaces, with coefficient depending on the Laplace transform parameter; a similar assumption also applies to probability fluxes. Such boundary conditions (or boundary conditions equivalent to them) are most commonly used when considering the diffusion in a membrane system. There is derived the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying the Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation. If this equation is not met, the Markov property is broken. In particular, it has been shown that the Markov property is broken for the system with one-sided fully permeable membrane and with a partially absorbing membrane. When a probability flux is continuous at the membrane, the general form of the boundary condition for which the fundamental solutions meet the BSCK equation is derived.