Boundary Homogenization for Target Search Problems

In this chapter, we describe several approximations in the theory of Laplacian transport near complex or heterogeneously reactive boundaries. This phe-nomenon, governed by the Laplace operator, is ubiquitous in fields as diverse as chemical physics, hydrodynamics, electrochemistry, heat transfer, wave propaga-tion, 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.