The dichotomous acceleration process in one dimension: position fluctuations

Abstract

We study the motion of a one-dimensional particle that reverses its direction of acceleration stochastically. We focus on two contrasting scenarios, where the waiting times between two consecutive acceleration reversals are drawn from (i) an exponential distribution and (ii) a power-law distribution ρ(τ ) ∼ τ−(1+α). We compute the mean, variance and short-time distribution of the position x (t) using a trajectory-based approach. We show that, while for the exponential waiting time,⟨x2(t)⟩ ∼ t3 at long times, for the power-law case, a non-trivial algebraic growth ⟨x2(t)⟩ ∼ t2ϕ(α) emerges, where ϕ(α) = 2, (5−α)/2 and 3/2 for α < 1, 1 < α ⩽ 2 and α>2, respectively. Interestingly, we find that the long-time position distribution in case (ii) is a function of the scaled variable x/tϕ(α) with an α-dependent scaling function, which has qualitatively very different shapes for α<1 and α>1. In contrast, for case (i), the typical long-time fluctuations of position are Gaussian.