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

Abstract

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 combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or non-local processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.

10 pages, 3 figures

References in corpus (13)

• Anomalous transport in the crowded world of biological cells
• Assessing non-Markovian dynamics
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• On measures of non-Markovianity: divisibility vs. backflow of information
• Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
• Fractional diffusion modeling of ion channel gating
• Paradoxes of Subdiffusive Infiltration in Disordered Systems
• Stochastic jump processes for non-Markovian quantum dynamics
• A Random Walk to a Non-Ergodic Equilibrium Concept
• Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File
• Hyperbolic subdiffusive impedance
• Linear Response and Fluctuation Dissipation Theorem for non-Poissonian Renewal Processes
• Random walk model of subdiffusion in a system with a thin membrane

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