# Fluctuating hydrodynamics and the Brownian motion of an active colloid near a wall

## TL;DR

This paper systematically derives the fluctuating hydrodynamics of an active colloid near a wall, revealing activity-induced breakdown of detailed balance and providing tools for Brownian dynamics simulations.

## Contribution

It introduces a systematic derivation of passive, active, and Brownian traction contributions for active colloids, including fluctuation-dissipation relations and a regularization method for simulations.

## Key findings

- Derived exact relations between Brownian traction and thermal forces.
- Showed the stationary distribution is non-Gibbsian near a wall.
- Provided a regularization for friction tensors ensuring positive variances.

## Abstract

The traction on the surface of a spherical active colloid in a thermally fluctuating Stokesian fluid contains passive, active, and Brownian contributions. Here we derive these three parts systematically, by "projecting out" the fluid using the boundary-domain integral representation of slow viscous flow. We find an exact relation between the statistics of the Brownian traction and the thermal forces in the fluid and derive, thereby, fluctuation-dissipation relations for every irreducible tensorial harmonic traction mode. The first two modes give the Brownian force and torque, from which we construct the Langevin and Smoluchowski equations for the position and orientation of the colloid. We emphasize the activity-induced breakdown of detailed balance and provide a prescription for computing the configuration-dependent variances of the Brownian force and torque. We apply these general results to an active colloid near a plane wall, the simplest geometry with configuration-dependent variances, and show that the stationary distribution is non-Gibbsian. We derive a regularization of the translational and rotational friction tensors, necessary for Brownian dynamics simulations, that ensures positive variances at all distances from the wall. The many-body generalization of these results is indicated.