# Steady State of an Active Brownian Particle in Two-Dimensional Harmonic Trap

## TL;DR

This paper derives an exact series solution for the steady-state distribution of a 2D active Brownian particle in a harmonic trap, revealing a re-entrant transition between active and passive regimes validated by simulations.

## Contribution

It provides the first exact series solution for the steady-state distribution of an active Brownian particle in a harmonic trap, enabling detailed analysis of different regimes.

## Key findings

- Identification of active and passive regimes
- Prediction of a re-entrant active-passive transition
- Validation through numerical simulations

## Abstract

We find an exact series solution for the steady-state probability distribution of a harmonically trapped active Brownian particle in two dimensions, in the presence of translational diffusion. This series solution allows us to efficiently explore the behavior of the system in different parameter regimes. Identifying "active" and "passive" regimes, we predict a surprising re-entrant active-to-passive transition with increasing trap stiffness. Our numerical simulations validate this finding. We discuss various interesting limiting cases wherein closed form expressions for the distributions can be obtained.