Shape matters: A Brownian microswimmer in a channel

TL;DR

This paper investigates how the shape and rotation center of a two-dimensional active Brownian particle influence its behavior in a channel, deriving equations for invariant density, reversal time, and effective diffusivity.

Contribution

It derives a reduced model for microswimmers in channels considering shape and rotation center effects, providing new formulas for invariant density, reversal time, and diffusivity.

Findings

Invariant density depends strongly on swimmer shape and rotation center.

Derived a formula for the mean reversal time of the swimmer.

Established that effective diffusivity is bounded by the mean reversal time.

Abstract

We consider the active Brownian particle (ABP) model for a two-dimensional microswimmer with fixed speed, whose direction of swimming changes according to a Brownian process. The probability density for the swimmer evolves according to a Fokker-Planck equation defined on the configuration space, whose structure depends on the swimmer's shape, center of rotation and domain of swimming. We enforce zero probability flux at the boundaries of configuration space. We derive a reduced equation for a swimmer in an infinite channel, in the limit of small rotational diffusivity, and find that the invariant density depends strongly on the swimmer's precise shape and center of rotation. We also give a formula for the mean reversal time: the expected time taken for a swimmer to completely reverse direction in the channel. Using homogenization theory, we find an expression for the effective longitudinal diffusivity of a swimmer in the channel, and show that it is bounded by the mean reversal time.