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

We consider an active Brownian particle in a d-dimensional harmonic trap, in the presence of translational diffusion. While the Fokker-Planck equation cannot 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 an 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 recently observed active-to-passive re-entrant behavior.

Automated summary

By studying active Brownian particles within a harmonic trap, the exact time dependence of moments can be evaluated using a Laplace transform approach. The kurtosis of displacement is calculated, showing differences between active steady state properties and equilibrium Gaussian form. The kurtosis increases with activity to asymptotic saturation, but varies non-monotonically with trap-stiffness, capturing observed active-to-passive re-entrant behavior.