Dichotomous acceleration process

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 with decay exponent $\alpha$. 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, $\langle x^2(t)\rangle \sim t^3$ at long times, for the power-law case, a non-trivial algebraic growth $\langle x^2(t)\rangle \sim t^{2\phi(\alpha)}$ emerges, where $\phi(\alpha)=2$, $(5-\alpha)/2,$ and $3/2$ for $\alpha<1,~1<\alpha\leq 2$ and $\alpha>2$, respectively. Interestingly, we find that the long-time position distribution in case (ii) is a function of the scaled variable $x/t^{\phi(\alpha)}$ 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.