Active ideal sedimentation: Exact two-dimensional steady states

TL;DR

This paper analytically solves the steady-state distribution of active Brownian particles under gravity in two dimensions, revealing two distinct spatial regimes as the Peclet number increases, characterized by different orientation and density profiles.

Contribution

It provides an exact analytical solution to the steady states of active particles under gravity, incorporating boundary conditions and revealing new spatial regimes.

Findings

Identification of two spatial regimes with increasing Peclet number

Different mean particle orientations in each regime

Variation of density profiles with height

Abstract

We consider an ideal gas of active Brownian particles that undergo self-propelled motion and both translational and rotational diffusion under the influence of gravity. We solve analytically the corresponding Smoluchowski equation in two space dimensions for steady states. The resulting one-body density is given as a series, where each term is a product of an orientation-dependent Mathieu function and a height-dependent exponential. A lower hard wall is implemented as a no-flux boundary condition. Numerical evaluation of the suitably truncated analytical solution shows the formation of two different spatial regimes upon increasing Peclet number. These regimes differ in their mean particle orientation and in their variation of the orientation-averaged density with height.