Effective diffusivity of a microswimmer in a lattice

Microswimmers display various patterns when they swim in complex environments, and their trajectories are studied both experimentally and theoretically for a wide range of geometries. In recent years, confinement in gel structure receives a great amount of attentions where swimmers exhibite a diffusive behavior in long time scales. In order to understand the effective diffusivity, we consider a 2-D active Brownian swimmer navigating through homogenous gel media. The swimmer travels with constant speed and its direction of swimming changes according to a Brownian process. The gel media are modelled by a periodic lattice grid, whose dimensions are large compared to the size of the swimmer. In this case, hydrodynamic interactions with boundaries are negligible since boundaries are lattice points. Steric interactions on the other hand, play an important role and they depend on a swimmer's shape. To avoid making ad-hoc assumptions on the swimmer's behavior near the lattice points, we only assume a swimmer can not penetrate lattice points and absorb the shape into a no-flux boundary condition. The invariant density of the swimmer then satisfies a Fokker--Planck equation, and we solve the effective diffusivity tensor in space asymptotically.