Diffusion in a Half-Space: From Lord Kelvin to Path Integrals

TL;DR

This paper explores diffusion in a half-space through various methods like diffusion equations, lattice walks, and path integrals, highlighting universal effects of geometrical constraints on transport phenomena.

Contribution

It provides a comparative analysis of different approaches to model confined diffusion and discusses potential generalizations of these methods.

Findings

Universal scaling relations identified for confined diffusion

Path integral approach offers new insights into boundary effects

Comparison of methods enhances understanding of transport in constrained geometries

Abstract

Many important transport phenomena are described by simple mathematical models rooted in the diffusion equation. Geometrical constraints present in such phenomena often have influence of a universal sort and manifest themselves in scaling relations and stable distribution functions. In this paper, I present a treatment of a random walk confined to a half--space using a number of different approaches: diffusion equations, lattice walks and path integrals. Potential generalizations are discussed critically.