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Abridged abstract: Inert interactions between randomly moving entities andspatial disorder play a crucial role in quantifying the diffusive properties ofa system. These interactions affect only the movement of the entities, andexamples range from molecules advancing along dendritic spines to anti-predatordisplacements of animals due to sparse vegetation. Despite the prevalence ofsuch systems, a general framework to model the movement explicitly in thepresence of spatial heterogeneities is missing. Here, we tackle this challengeand develop an analytic theory to model inert particle-environment interactionsin domains of arbitrary shape and dimensions. We use a discrete spaceformulation which allows us to model the interactions between an agent and theenvironment as perturbed dynamics between lattice sites. Interactions fromspatial disorder, such as impenetrable and permeable obstacles or regions ofincreased or decreased diffusivity, as well as many others, can be modelledusing our framework. We provide exact expressions for the generating functionof the occupation probability of the diffusing particle and related transportquantities such as first-passage, return and exit probabilities and theirrespective means. We uncover a surprising property, the disorder indifferencephenomenon of the mean first-passage time in the presence of a permeablebarrier in quasi-1D systems. We demonstrate the widespread applicability of ourformalism by considering three examples that span across scales anddisciplines. (1) We explore an enhancement strategy of transdermal drugdelivery. (2) We associate the disorder with a decision-making process of ananimal to study thigmotaxis. (3) We illustrate the use of spatialheterogeneities to model inert interactions between particles by modelling thesearch for a promoter region on the DNA by transcription factors during genetranscription.

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