Active particle in one dimension subjected to resetting with memory

TL;DR

This paper investigates a one-dimensional active particle with memory-based resetting, deriving exact position distributions, analyzing the transition from ballistic to logarithmic diffusion, and establishing a large deviation principle with unique time-scaling.

Contribution

It introduces an exact analytical framework for an active particle with memory-based resetting, combining activity and non-Markovian effects.

Findings

Derived exact Fourier space position distribution.

Identified crossover from ballistic to logarithmic diffusion.

Established large deviation principle with logarithmic time-scaling.

Abstract

The study of diffusion with preferential returns to places visited in the past has attracted an increased attention in recent years. In these highly non-Markov processes, a standard diffusive particle intermittently resets at a given rate to previously visited positions. At each reset, a position to be revisited is randomly chosen with a probability proportional to the accumulated amount of time spent by the particle at that position. These preferential revisits typically generate a very slow diffusion, logarithmic in time, but still with a Gaussian position distribution at late times. Here we consider an active version of this model, where between resets the particle is self-propelled with constant speed and switches direction in one dimension according to a telegraphic noise. Hence there are two sources of non-Markovianity in the problem. We exactly derive the position distribution in Fourier space, as well as the variance of the position at all times. The crossover from the short-time ballistic regime, dominated by activity, to the large-time anomalous logarithmic growth induced by memory is studied. We also analytically derive a large deviation principle for the position, which exhibits a logarithmic time-scaling instead of the usual algebraic form. Interestingly, at large distances, the large deviations become independent of time and match the non-equilibrium steady state of a particle under resetting to its starting position only.