Stationary fluctuations of run-and-tumble particles

TL;DR

This paper analyzes the stationary fluctuations of run-and-tumble particles, demonstrating convergence to Ornstein-Uhlenbeck processes and deriving covariance and large deviation properties, including effects of interactions and memory.

Contribution

It introduces a rigorous analysis of stationary fluctuations for both independent and interacting run-and-tumble particles, including non-Markovian effects and large deviation rate functions.

Findings

Joint densities converge to an infinite-dimensional Ornstein-Uhlenbeck process.

Total density fluctuations form a non-Markovian Gaussian process with explicit covariance.

Large deviation rate functions include memory effects in small noise limits.

Abstract

We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.