Diffusive-to-Ballistic transition in a Persistent Random Walk

TL;DR

This paper investigates a dynamical transition in persistent random walks with time-dependent reversal probabilities, identifying a critical point at which the system shifts from super-diffusive to ballistic behavior.

Contribution

It introduces a criterion for a non-equilibrium transition in persistent random walks with time-dependent velocities, supported by detailed analysis and generalization to various reversal probabilities.

Findings

Transition at α=1 separates super-diffusive and ballistic regimes.

The transition is characterized by velocity correlations and displacement fluctuations.

The transition persists across different spatial dimensions under isotropy.

Abstract

We study persistent random walk with time dependent velocity reversal probabilities and identify a criterion for a non-equilibrium dynamical transition. As a representative example, we consider a power law reversal probability $p(t)\sim t^{-\alpha}$ and show that the system undergoes a transition at $\alpha=1$, separating a super-diffusive regime for $\alpha<1$ from ballistic regime for $\alpha \geq 1$. Using the results for velocity correlations and persistence statistics, together with finite time scaling of the Binder cumulant and displacement fluctuations, we characterize the transition and its properties in detail. We further argue that the transition is not limited to the power law form, but can also arise for several other time dependent reversal probabilities satisfying the same criterion. The transition persists in arbitrary spatial dimensions provided isotropy of the velocity space is preserved.