Dynamics of lattice random walk within regions composed of different media and interfaces

TL;DR

This paper analyzes the dynamics of lattice random walks in heterogeneous media separated by an interface, providing exact solutions for propagators, boundary conditions, and first-passage times for different interface placements and boundary conditions.

Contribution

It introduces two models for random walk interfaces, derives exact propagators and boundary conditions, and compares their steady-state and first-passage properties.

Findings

Exact propagators for both interface types and boundary conditions.

Steady-state probability is step-like for Type A and uniform for Type B.

Derived explicit expressions for first-passage times.

Abstract

We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (1) Type A dynamics whereby the interface is placed between two lattice points, and (2) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green's function or propagator for the composite system in unbounded domain as well as domains confined with reflecting, absorbing, and mixed boundaries. For the case with reflecting confinement in the absence of bias, the steady-state probability shows a step-like behavior for the Type A dynamics, while it is uniform for the Type B dynamics. We also derive explicit expressions for the first-passage probability and the mean first-passage time, and compare the hitting time dependence to a single target. Finally, considering the continuous-space continuous-time limit of the propagator, we obtain the boundary conditions at the interface. At the interface, while the flux is the same, the probability density is discontinuous for Type A and is continuous for Type B. For the latter we derive a generalized version of the so-called leather boundary condition in the appropriate limit.