Run-and-tumble motion: the role of reversibility

TL;DR

This paper analyzes a model of active particles with internal states, showing that reversibility maximizes active diffusion and deriving explicit formulas and large deviations principles for the system.

Contribution

It demonstrates that the active diffusion coefficient is maximized for reversible internal state processes and provides explicit calculations and large deviations results.

Findings

Active diffusion is maximized for reversible state processes.

Explicit formulas for two-state systems are derived.

Large deviations principles are established for the model.

Abstract

We study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the `active part' of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier-Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.