Particle-Environment Interactions In Arbitrary Dimensions: A Unifying Analytic Framework To Model Diffusion With Inert Spatial Heterogeneities

TL;DR

This paper develops a comprehensive analytic framework to model particle diffusion in complex environments with spatial heterogeneities across arbitrary dimensions, unifying various scenarios like obstacles, diffusivity changes, and biological processes.

Contribution

The authors introduce a discrete space formalism that explicitly models inert particle-environment interactions in arbitrary-shaped domains, providing exact expressions for key transport quantities.

Findings

Exact formulas for occupation probabilities and transport metrics.

Discovery of disorder indifference phenomenon in mean first-passage time.

Applicability demonstrated across biological, ecological, and physical systems.

Abstract

Abridged abstract: Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system. These interactions affect only the movement of the entities, and examples range from molecules advancing along dendritic spines to anti-predator displacements of animals due to sparse vegetation. Despite the prevalence of such systems, a general framework to model the movement explicitly in the presence of spatial heterogeneities is missing. Here, we tackle this challenge and develop an analytic theory to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We use a discrete space formulation which allows us to model the interactions between an agent and the environment as perturbed dynamics between lattice sites. Interactions from spatial disorder, such as impenetrable and permeable obstacles or regions of increased or decreased diffusivity, as well as many others, can be modelled using our framework. We provide exact expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return and exit probabilities and their respective means. We uncover a surprising property, the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi-1D systems. We demonstrate the widespread applicability of our formalism by considering three examples that span across scales and disciplines. (1) We explore an enhancement strategy of transdermal drug delivery. (2) We associate the disorder with a decision-making process of an animal to study thigmotaxis. (3) We illustrate the use of spatial heterogeneities to model inert interactions between particles by modelling the search for a promoter region on the DNA by transcription factors during gene transcription.