Active Brownian particle in harmonic trap: exact computation of moments, and re-entrant transition

TL;DR

This paper provides an exact method to compute moments of an active Brownian particle in a harmonic trap, revealing non-monotonic behavior and re-entrant transitions in its steady-state properties.

Contribution

It introduces a Laplace transform approach to exactly evaluate moments of the particle's distribution, capturing re-entrant transitions in active matter systems.

Findings

Exact moments of the particle's position and orientation are computed.

Kurtosis of displacement shows non-monotonic variation with trap stiffness.

Re-entrant transition behavior is observed in the steady state.

Abstract

We consider an active Brownian particle in a $d$-dimensional harmonic trap, in the presence of translational diffusion. While the Fokker-Planck equation can not in general be solved to obtain a closed form solution of the joint distribution of positions and orientations, as we show, it can be utilized to evaluate the exact time dependence of all moments, using a Laplace transform approach. We present explicit calculation of several such moments at arbitrary times and their evolution to the steady state. In particular we compute the kurtosis of the displacement, a quantity which clearly shows the difference of the active steady state properties from the equilibrium Gaussian form. We find that it increases with activity to asymptotic saturation, but varies non-monotonically with the trap-stiffness, thereby capturing a recently observed active- to- passive re-entrant behavior.