Boundary Integral Methods for Particle Diffusion in Complex Geometries: Shielding, Confinement, and Escape

TL;DR

This paper introduces a boundary integral numerical method using Laplace transforms to efficiently solve diffusion problems in complex, unbounded geometries, capturing effects like shielding, confinement, and targeted escape.

Contribution

The paper presents a novel boundary integral approach with Laplace transform techniques for diffusion in complex geometries, enabling long timescale analysis without traditional time-stepping.

Findings

Demonstrates shielding effects in complex geometries.

Shows geometry-guided diffusion to specific sites.

Illustrates escape routes in maze-like structures.

Abstract

We present a numerical method for the solution of diffusion problems in unbounded planar regions with complex geometries of absorbing and reflecting bodies. Our numerical method applies the Laplace transform to the parabolic problem, yielding a modified Helmholtz equation which is solved with a boundary integral method. Returning to the time domain is achieved by quadrature of the inverse Laplace transform by deforming along the so-called Talbot contour. We demonstrate the method for various complex geometries formed by disjoint bodies of arbitrary shape on which either uniform Dirichlet or Neumann boundary conditions are applied. The use of the Laplace transform bypasses constraints with traditional time-stepping methods and allows for integration over the long equilibration timescales present in diffusion problems in unbounded domains. Using this method, we demonstrate shielding effects where the complex geometry modulates the dynamics of capture to absorbing sets. In particular, we show examples where geometry can guide diffusion processes to particular absorbing sites, obscure absorbing sites from diffusing particles, and even find the exits of confining geometries, such as mazes.