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  • Letter
  • Open Access

Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics

Toby Kay1,* and Luca Giuggioli1,2,†

  • 1Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, United Kingdom
  • 2Bristol Centre for Complexity Sciences, University of Bristol Bristol, BS8 1UB, United Kingdom

  • *toby.kay@bristol.ac.uk
  • †Luca.Giuggioli@bristol.ac.uk
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Phys. Rev. Research 4, L032039 – Published 7 September, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L032039

Abstract

The diffusion equation is the primary tool to study the movement dynamics of a free Brownian particle, but when spatial heterogeneities in the form of permeable interfaces are present, no fundamental equation has been derived. Here we obtain such an equation from a microscopic description using a lattice random walk model. The sought after Fokker-Planck description and the corresponding backward Kolmogorov equation are employed to investigate first-passage and local time statistics and gain new insights. Among them a surprising phenomenon, in the case of a semibounded domain, is the appearance of a regime of dependence and independence on the location of the permeable barrier in the mean first-passage time. The new formalism is completely general: it allows to study the dynamics in the presence of multiple permeable barriers as well as reactive heterogeneities in bounded or unbounded domains and under the influence of external forces.

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Phys. Rev. Research 4, L032039– Published 7 September, 2022
  • Received 14 February 2022
  • Accepted 15 June 2022
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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