Exact moments and re-entrant transitions in the inertial dynamics of active Brownian particles

Manish Patel, Debasish Chaudhuri

Published 2023-10-01Version 1

In this study, we investigate the behavior of free inertial Active Brownian Particles (ABP) in the presence of thermal noise. While finding a closed-form solution for the joint distribution of positions, orientations, and velocities using the Fokker-Planck equation is generally challenging, we utilize a Laplace transform method to obtain the exact temporal evolution of all dynamical moments in arbitrary dimensions. Our expressions in $d$ dimensions reveal that inertia significantly impacts steady-state kinetic temperature and swim pressure while leaving the late-time diffusivity unchanged. Notably, the velocity distribution exhibits a remarkable re-entrant transition from passive Gaussian to active non-Gaussian behaviors. We construct a phase diagram using the exact expression of the $d$-dimensional kurtosis as a function of activity and inertia. Our analytic expressions describe steady states and offer insights into time-dependent crossovers observed in various velocity and displacement moments.