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-dimensionalactive Brownian particle trapped in an isotropic harmonic potential. Using the passive Brownian particle asbasis states we show that the Fokker-Planck operator becomes lower diagonal, implying that theeigenvalues are unaffected by the activity. The propagator is then expressed as a combination of theequilibrium eigenstates with weights obeying exact iterative relations. We show that for the low-ordercorrelation functions, such as the positional autocorrelation function, the recursion terminates at finite orderin the P ́eclet number, allowing us to generate exact compact expressions and derive the velocityautocorrelation function and the time-dependent diffusion coefficient. The nonmonotonic behavior of latterquantities serves as a fingerprint of the nonequilibrium dynamics.

The folder contains the codes for numerics and simulations, and for creating figures.