Different approach to the modeling of nonfree particle diffusion

TL;DR

This paper introduces a geometric graph-based approach to model nonfree particle diffusion, generalizing the central limit theorem to include partially absorbing barriers and providing new analytic expressions for the propagator.

Contribution

It presents a novel framework using geometric graphs to model nonfree diffusion and extends the central limit theorem to account for partial absorption in barriers.

Findings

Derived continuum-limit propagator as sum of Gaussians and plane waves

Unified traditional methods as special cases

Revealed phenomena like line splitting and band gaps due to barriers

Abstract

A new approach to the modeling of nonfree particle diffusion is presented. The approach uses a general setup based on geometric graphs (networks of curves), which means that particle diffusion in anything from arrays of barriers and pore networks to general geometric domains can be considered and that the (free random walk) central limit theorem can be generalized to cover also the nonfree case. The latter gives rise to a continuum-limit description of the diffusive motion where the effect of partially absorbing barriers is accounted for in a natural and non-Markovian way that, in contrast to the traditional approach, quantifies the absorptivity of a barrier in terms of a dimensionless parameter in the range 0 to 1. The generalized theorem gives two general analytic expressions for the continuum-limit propagator: an infinite sum of Gaussians and an infinite sum of plane waves. These expressions entail the known method-of-images and Laplace eigenfunction expansions as special cases and show how the presence of partially absorbing barriers can lead to phenomena such as line splitting and band gap formation in the plane wave wave-number spectrum.