Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions

TL;DR

This paper develops an analytical framework for biased lattice random walks confined within arbitrary boundaries, deriving explicit solutions for propagators and first-passage times in multiple dimensions, revealing complex dynamic behaviors.

Contribution

It provides the first analytical solutions for confined biased lattice random walks in arbitrary dimensions, including propagators and first-passage time calculations, previously accessible only through computational methods.

Findings

Surprising saddle points in propagator dynamics with reflecting boundaries

Bimodal first-passage probability distributions in periodic domains

Optimal target placement near boundaries for shorter mean first-passage times

Abstract

Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a $d$-dimensional box, $d$-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatio-temporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimisation of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multi-target environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them.