Passive and active particles in a lattice of obstacles

Microswimmers and passive particles frequently encounter obstacles that slow down their progress. We consider a regular lattice of small obstacles, with which the particles interact only sterically. A natural question is: as they bounce around the lattice, what is the effective diffusion coefficient for the particles? We show that Brownian passive particles obey the same equation as for Rayleigh's problem of conduction in a perforated solid, with a similar solution in terms of reflections. For active particles, a homogenization approach involves a novel cell problem with boundary layers around columnar obstacles.