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

TL;DR

This paper derives exact dynamical moments for inertial active Brownian particles, revealing how inertia influences steady-state properties, velocity distributions, and phase transitions, with implications for related active particle models.

Contribution

It provides the first exact expressions for moments and phase diagrams of inertial ABPs, highlighting inertia's role in non-Gaussian behaviors and re-entrant transitions.

Findings

Inertia affects steady-state kinetic temperature and swim pressure.

Velocity distribution shows re-entrant transition from Gaussian to non-Gaussian.

Exact phase diagram based on kurtosis captures transition behaviors.

Abstract

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, as a function of activity and inertia, the steady-state velocity distribution exhibits a remarkable re-entrant crossover from passive Gaussian to active non-Gaussian behaviors. We construct a corresponding phase diagram using the exact expression of the $d$-dimensional kurtosis. Our analytic expressions describe steady states and offer insights into time-dependent crossovers observed in moments of velocity and displacement. Our calculations can be extended to predict up to second-order moments for run-and-tumble particles (RTP) and the active Ornstein-Uhlenbeck process (AOUP). Additionally, the kurtosis shows differences from AOUP.