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

Abstract

In this study, we investigate the behavior of free inertial active Brownian particles 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 and the active Ornstein–Uhlenbeck process (AOUP). Additionally, the kurtosis shows differences from AOUP.

Exact moments for trapped active particles: inertial impact on steady-state properties and re-entrance

“…In the vanishing β limit, the expression agrees with the earlier estimate for free iABPs [65]. In the overdamped limit M → 0, the dimensionless k B T kin = 1, set by translational diffusivity.…”

“…In the steady state, the mean position also vanishes ⟨r⟩ st = 0 and shows an oscillatory evolution towards the steady state with frequency ω/M if βM > 1/4, as for mean velocity. In the limit of vanishing inertia, equation (11) restores the previously known result of overdamped trapped ABPs [62], and in the vanishing trap strength, it restores the result of free iABPs [65].…”

“…In the vanishing trap strength limit, ⟨v ∥ ⟩ st takes the previously known form of free iABPs [65]. In the other limit of β → ∞, both projections ⟨r ∥ ⟩ st and ⟨v ∥ ⟩ st vanishes.…”

“…The case of 4βM = 1 corresponds to critical damping in oscillations. In the limit of vanishing trap strength, equation ( 9) reduces to the known expression for free iABPs [65].…”

“…Through a Laplace transform of the governing Fokker-Planck equation, we demonstrate the precise time evolution of all moments of any desired dynamical variable in arbitrary d dimensions, a primary accomplishment of this work. This method, initially devised for investigating worm-like chains in 1952 [61], has been recently adapted to study ABP dynamics [16,[62][63][64][65]. We provide the exact expressions for the time evolution of various position and velocity moments, validating them against direct numerical simulations.…”