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

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.

Figures (6)

• Figure 1: The SCGF $\Lambda (s)$ is plotted as a function the biasing field $s$. The black dashed line is $\Lambda(s)=\ln (p e^s+q e^{-s})$. The red line is the solution of (\ref{['TE']}). The blue dotted line is the result of the stochastic cloning simulation where the number of clones is $10^5$ and the run time is $500000$. The parameters are $p = 0.6,\ \ q = 0.4,\ \ b = 1.0,\ \ a = 1.5$ (left) and $a = 0.6$ (right). The vertical lines are the locations of the transition points $s_{c}^{(1)} = - 0.405$ and $s_{c}^{(2)}=0$.
• Figure 2: The rate function $I(v)$ (left) and the probability distribution function $P(v)$ (right) are plotted as a function of $v$ for $p = 0.6$ and $t = 10$.
• Figure 3: Mean current $\langle v \rangle$ at $s=0$ as a function of the aging strength $a$ for $p=0.6$ and $b=1.0$. The dashed vertical line at $a=1$ indicates the transition between the unbound regime (where the current is constant at $p-q$) and the bound regime (where resets reduce the current). This marks a continuous (second-order) transition in the parameter space. The dotted line is obtained from Monte Carlo simulations averaging over $60$ samples. The total time consists of $10^6$ steps.
• Figure 4: The plot of the SCGF $\Lambda(k)$ for two values of the biasing field $s$: $s = + 0.2$ (left) and $s$=-0.2 (right). See the text for more detail.
• Figure 5: The plot of the SCGF $\Lambda (s)$ as a function of $s$. The vertical lines are $s_{c}^{(1)}$ and $s_{c}^{(2)}$ given by (\ref{['roots']}). For $p>q$ the blue curve is always bellow the red and the black curves; therefore, does not contribute in $\Lambda (s)$.