Theory for the Casimir effect and the partitioning of active matter

Active Brownian particles (ABPs) distribute non-homogeneously near surfaces, and understanding how this depends on system properties-size, shape, activity level, etc.-is essential for predicting and exploiting the behavior of active matter systems. Active particles accumulate at no-flux surfaces owing to their persistent swimming, which depends on their intrinsic swim speed and reorientation time, and are subject to confinement effects when their run or persistence length is comparable to the characteristic size of the confining geometry. It has been observed in simulations that two parallel plates experience a "Casimir effect" and attract each other when placed in a dilute bath of ABPs. In this work, we provide a theoretical model based on the Smoluchowski equation and a macroscopic mechanical momentum balance to analytically predict this attractive force. We extend this method to describe the concentration partitioning of active particles between a confining channel and a reservoir, showing that the ratio of the concentration in the channel to that in the bulk increases as either run length increases or channel height decreases. The theoretical results agree well with Brownian dynamics simulations and finite element calculations.