Kinematics of Persistent Random Walkers with Two Distinct Modes of Motion

TL;DR

This paper develops an analytical framework to understand the motion of active particles switching between two modes, linking microscopic properties to macroscopic transport behavior, with explicit results for diffusion and displacement over time.

Contribution

It introduces a formalism that connects microscopic switching dynamics and persistence to macroscopic transport metrics for particles with two motion modes.

Findings

Derived analytical expressions for initial anomalous exponent and crossover time.

Obtained exact formula for the evolution of mean square displacement.

Provided a method to optimize transport properties based on particle dynamics.

Abstract

We study the stochastic motion of active particles that undergo spontaneous transitions between two distinct modes of motion. Each mode is characterized by a velocity distribution and an arbitrary (anti-)persistence. We present an analytical formalism to provide a quantitative link between these two microscopic statistical properties of the trajectory and macroscopically observable transport quantities of interest. For exponentially distributed residence times in each state, we derive analytical expressions for the initial anomalous exponent, the characteristic crossover time to the asymptotic diffusive dynamics, and the long-term diffusion constant. We also obtain an exact expression for the time evolution of the mean square displacement over all time scales and provide a recipe to obtain higher displacement moments. Our approach enables us to disentangle the combined effects of velocity, persistence, and switching probabilities between the two states on the kinematics of particles in a wide range of stochastic active/passive processes and to optimize the transport quantities of interest with respect to any of the particle dynamics properties.