Analytic Solution of an Active Brownian Particle in a Harmonic Well

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 nonequilibrium dynamics.

Automated summary

This study provides an analytical solution for the time-dependent Fokker-Planck equation of a two-dimensional active Brownian particle. By using the passive Brownian particle as basis states, the authors show that the Fokker-Planck operator becomes lower diagonal, indicating that the eigenvalues are not affected by activity. They further express the propagator as a combination of equilibrium eigenstates with exact iterative relations for the weights.