Semi-Markovian Dynamics of a Self-Propelled Particle in a Confined Environment: A Large-Deviation Study

TL;DR

This paper investigates large deviations in the velocity fluctuations of a self-propelled particle confined within an environment, modeling its dynamics as a semi-Markov process with phase transitions and analyzing the resulting dynamical phase transitions.

Contribution

It introduces a semi-Markovian model for self-propelled particles with phase-dependent dynamics and characterizes the conditions for dynamical phase transitions in velocity fluctuations.

Findings

Identifies conditions for first- and second-order dynamical phase transitions.

Demonstrates the impact of aging strength on the nature of phase transitions.

Validates analytical predictions with computer simulations.

Abstract

We study the large deviations of the time-integrated current for a self-propelled particle moving within a confined environment. The dynamics is modeled as a semi-Markovian process, where the transitions between a \textit{normal running phase} (Phase $0$) and a \textit{wall-attached phase} (Phase $1$) are governed by time-dependent reset probabilities. We study two different examples: In the first case, the particle undergoes a biased random walk in Phase $0$, while it intermittently resets and interacts with the container boundaries, remaining stationary in Phase $1$. In this scenario, the reset probabilities for transitions between the two phases follow an ``aging'' logic. In the second case, the particle alternates between two active phases: a Markovian Phase $0$ characterized by memoryless, downstream-biased motion, and a semi-Markovian Phase $1$ with a reversed, upstream bias representing boundary-attached navigation. Here, we assume a time-independent survival probability in Phase $0$ and a time-dependent one in Phase $1$. By analyzing the Scaled Cumulant Generating Function (SCGF) in the long-time limit, we derive the conditions for Dynamical Phase Transition (DPT)s in the fluctuations of the particle velocity. We demonstrate that, depending on the aging strength, the system exhibits either discontinuous (first-order) or continuous (second-order) DPTs. Analytical predictions are validated via computer simulations.