A Geometrical Method for the Smoluchowski Equation on the Sphere

TL;DR

This paper introduces a simple, geometry-based numerical algorithm for simulating Brownian particle diffusion on a sphere under external potential, effectively capturing curvature effects in both dynamic and stationary states.

Contribution

The authors develop an efficient, algebraic, geometry-based numerical method that converges to solutions of the Smoluchowski equation on the sphere, accounting for curvature effects.

Findings

Algorithm effectively simulates particle trajectories on the sphere.

Curvature effects are incorporated into both time-dependent and stationary solutions.

Method is simple, efficient, and converges in the weak sense.

Abstract

A study of the diffusion of a passive Brownian particle on the surface of a sphere and subject to the effects of an external potential, coupled linearly to the probability density of the particle's position, is presented through a numerical algorithm devised to simulate the trajectories of an ensemble of Brownian particles. The algorithm is based on elementary geometry and practically only algebraic operations are used, which makes the algorithm efficient and simple, and converges, in the \textit{weak sense}, to the solutions of the Smoluchowski equation on the sphere. Our findings show that the global effects of curvature are taken into account in both the time dependent and stationary processes.