New homogenization approaches for stochastic transport through heterogeneous media

TL;DR

This paper introduces new homogenization methods for stochastic transport in layered heterogeneous media, deriving effective coefficients that account for internal interfaces and bias, significantly improving upon traditional approaches.

Contribution

The authors develop a novel class of homogenization approximations for stochastic transport in layered media, providing explicit formulas for effective coefficients and internal interface handling.

Findings

Effective transport coefficients derived for layered media

Internal interfaces influence overall transport bias

Generalized approach yields different results than traditional methods

Abstract

The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an $\textit{effective}$ homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the $k$th moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers gives rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Example calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.