Confined run and tumble particles with non-Markovian tumbling statistics

Abstract

Confined active particles constitute simple, yet realistic, examples of systems that converge into a non-equilibrium steady state. We investigate a run-and-tumble particle in one spatial dimension, trapped by an external potential, with a given distribution $g(t)$ of waiting times between tumbling events whose mean value is equal to $\tau$. Unless $g(t)$ is an exponential distribution (corresponding to a constant tumbling rate), the process is non-Markovian, which makes the analysis of the model particularly challenging. We use an analytical framework involving effective position-dependent tumbling rates, to develop a numerical method that yields the full steady-state distribution (SSD) of the particle's position. The method is very efficient and requires modest computing resources, including in the large-deviations and/or small-$\tau$ regime, where the SSD can be related to the the large-deviation function, $s(x)$, via the scaling relation $P_{\mathrm{st}}(x)\sim e^{-s\left(x\right)/\tau }$.

11 pages, 6 figures

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• Towards neural reinforcement learning for large deviations in nonequilibrium systems with memory
• Semi-Markovian switching in a fluctuating harmonic trap: An age-structured formulation