Active Brownian Motion with Directional Reversals

Abstract

Active Brownian motion with intermittent direction reversals are common in a class of bacteria like {\it Myxococcus xanthus} and {\it Pseudomonas putida}. We show that, for such a motion in two dimensions, the presence of the two time scales set by the rotational diffusion constant $D_R$ and the reversal rate $\gamma$ gives rise to four distinct dynamical regimes: (I) $t\ll \min ({\gamma }^{-1},D_R^{-1}),$ (II) ${\gamma }^{-1}\ll t\ll D_R^{-1}$, (III) $D_R^{-1}\ll t\ll {\gamma }^{-1}$, and (IV) $t\gg \max ({\gamma }^{-1}$, $D_R^{-1})$, showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly non-diffusive and anisotropic behavior at short-times to a diffusive isotropic behavior via an intermediate regime (II) or (III). In regime (II), we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable $z\propto x_{\perp }/t$ with a non-trivial scaling function, $f(z)=(2{\pi }^3{)}^{-1/2}\Gamma (1/4+iz)\Gamma (1/4-iz)$. Furthermore, by computing the exact first-passage time distribution, we show that a novel persistence exponent $\alpha =1$ emerges due to the direction reversal in this regime.