Exact computation of the mean velocity, molecular diffusivity and dispersivity of a particle moving on a periodic lattice

TL;DR

This paper presents an exact analytical method for calculating the long-term velocity and dispersivity of a particle performing a biased random walk on a periodic lattice with obstacles, providing an alternative to numerical simulations.

Contribution

It introduces a novel analytical scheme for computing mean velocity and dispersivity on periodic lattices, applicable to complex trapping geometries and validated against existing zero-field results.

Findings

Exact dispersivity calculations for obstacle-free lattices.

Agreement with zero-field results from previous studies.

Method applicable to complex lattice structures and trapping geometries.

Abstract

A straightforward analytical scheme is proposed for computing the long-time, asymptotic mean velocity and dispersivity (effective diffusivity) of a particle undergoing a discrete biased random walk on a periodic lattice amongst an array of immobile, impenetrable obstacles. The results of this Taylor-Aris dispersion-based theory are exact, at least in an asymptotic sense, and furnish an analytical alternative to conventional numerical lattice Monte Carlo simulation techniques. Results obtained for an obstacle-free lattice are employed to establish generic relationships between the transition probabilities, lattice size and jump time. As an example, the dispersivity is computed for a solute moving through an isotropic array of obstacles under the influence of a finite external field. The calculation scheme is also shown to agree with existing zero-field results, the latter obtained elsewhere either by first-passage time analysis or use of the Nernst-Einstein equation in the zero-field limit. The generality of this scheme permits the study of more complex lattice structures, in particular trapping geometries.