Analytic Solution of an Active Brownian Particle in a Harmonic Well

TL;DR

This paper derives an exact analytical solution for the dynamics of an active Brownian particle in a harmonic trap, revealing non-equilibrium features through explicit correlation functions and diffusion behavior.

Contribution

It provides a novel analytical framework using eigenstates to solve the Fokker-Planck equation for active particles in harmonic potentials, highlighting eigenvalue invariance and recursive solution methods.

Findings

Exact expressions for positional autocorrelation functions

Velocity autocorrelation function derived explicitly

Time-dependent diffusion coefficient exhibits nonmonotonic behavior

Abstract

We provide an analytical solution for the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle trapped in an isotropic harmonic potential. Using the passive Brownian particle as basis states we show that the Fokker-Planck operator becomes lower diagonal, implying that the eigenvalues are unaffected by the activity. The propagator is then expressed as a combination of the equilibrium eigenstates with weights obeying exact iterative relations. We show that for the low-order correlation functions, such as the positional autocorrelation function, the recursion terminates at finite order in the P\'eclet number allowing us to generate exact compact expressions and derive the velocity autocorrelation function and the time-dependent diffusion coefficient. The nonmonotonic behavior of latter quantities serves as a fingerprint of the non-equilibrium dynamics.