Field Theory of Active Brownian Particles in Potentials

TL;DR

This paper develops a field theory framework for Active Brownian Particles in 2D and 3D, enabling analytical calculations of key properties and setting the stage for studying interacting systems.

Contribution

It introduces a novel field theory approach for ABPs in multiple dimensions, allowing exact and perturbative analysis of their behavior in external potentials.

Findings

Exact mean squared displacement in 2D

Perturbative density in external potentials

Effective diffusion constant corrections

Abstract

The Active Brownian Particle (ABP) model exemplifies a wide class of active matter particles. In this work, we demonstrate how this model can be cast into a field theory in both two and three dimensions. Our aim is manifold: we wish both to extract useful features of the system, as well as to build a framework which can be used to study more complex systems involving ABPs, such as those involving interaction. Using the two-dimensional model as a template, we calculate the mean squared displacement exactly, and the one-point density in an external potential perturbatively. We show how the effective diffusion constant appears in the barometric density formula to leading order, and determine the corrections to it. We repeat the calculation in three dimensions, clearly a more challenging setup. Comparing different ways to capture the self-propulsion, we find that its perturbative treatment results in more tractable derivations without loss of exactness, where this is accessible.