Boundary homogenization for target search problems

TL;DR

This paper reviews mathematical techniques like boundary homogenization and effective medium approximation to simplify complex boundary problems in Laplacian transport, aiding in diverse scientific applications such as target search and first-passage time analysis.

Contribution

It provides a comprehensive overview of boundary homogenization methods and their applications across multiple fields, highlighting their advantages and limitations.

Findings

Effective boundary approximation simplifies complex geometries.

Homogenization techniques improve modeling efficiency.

Applications include target search and first-passage time analysis.

Abstract

In this review, we describe several approximations in the theory of Laplacian transport near complex or heterogeneously reactive boundaries. This phenomenon, governed by the Laplace operator, is ubiquitous in fields as diverse as chemical physics, hydrodynamics, electrochemistry, heat transfer, wave propagation, self-organization, biophysics, and target search. We overview the mathematical basis and various applications of the effective medium approximation and the related boundary homogenization when a complex heterogeneous boundary is replaced by an effective much simpler boundary. We also discuss the constant-flux approximation, the Fick-Jacobs equation, and other mathematical tools for studying the statistics of first-passage times to a target. Numerous examples and illustrations are provided to highlight the advantages and limitations of these approaches.