Direction reversing active Brownian particle in a harmonic potential.

Abstract

We study the two-dimensional motion of an active Brownian particle of speed <i>v</i>₀, with intermittent directional reversals in the presence of a harmonic trap of strength <i>μ</i>. The presence of the trap ensures that the position of the particle eventually reaches a steady state where it is bounded within a circular region of radius <i>v</i>₀/<i>μ</i>, centered at the minimum of the trap. Due to the interplay between the rotational diffusion constant <i>D</i>_R, reversal rate <i>γ</i>, and the trap strength <i>μ</i>, the steady state distribution shows four different types of shapes, which we refer to as active-I & II, and passive-I & II phases. In the active-I phase, the weight of the distribution is concentrated along an annular region close to the circular boundary, whereas in active-II, an additional central diverging peak appears giving rise to a Mexican hat-like shape of the distribution. The passive-I is marked by a single Boltzmann-like centrally peaked distribution in the large <i>D</i>_R limit. On the other hand, while the passive-II phase also shows a single central peak, it is distinguished from passive-I by a non-Boltzmann like divergence near the origin. We characterize these phases by calculating the exact analytical forms of the distributions in various limiting cases. In particular, we show that for <i>D</i>_R ≪ <i>γ</i>, the shape transition of the two-dimensional position distribution from active-II to passive-II occurs at <i>μ</i> = <i>γ</i>. We compliment these analytical results with numerical simulations beyond the limiting cases and obtain a qualitative phase diagram in the (<i>D</i>_R, <i>γ</i>, <i>μ</i>^-1) space.