Dichotomous acceleration process in one dimension: Position fluctuations

We study the motion of a one-dimensional particle which 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 $\rho (\tau )\sim {\tau }^{-(1+\alpha )}$. 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, $⟨x^2(t)⟩\sim t^3$ at long times, for the power-law case, a non-trivial algebraic growth $⟨x^2(t)⟩\sim t^{2\phi (\alpha )}$ emerges, where $\phi (\alpha )=2$, $(5-\alpha )\mathrm{/}2,$ and $3\mathrm{/}2$ for $\alpha <1,1<\alpha \le 2$ and $\alpha >2$, respectively. Interestingly, we find that the long-time position distribution in case (ii) is a function of the scaled variable $x\mathrm{/}{t}^{\phi (\alpha )}$ with an $\alpha$-dependent scaling function, which has qualitatively very different shapes for $\alpha <1$ and $\alpha >1$. In contrast, for case (i), the typical long-time fluctuations of position are Gaussian.