Toward the full short-time statistics of an active Brownian particle on the plane

Abstract

We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method (OFM) to study large deviations of the particle position coordinates $x$ and $y$. We determine the optimal paths of the ABP, conditioned on reaching specified values of $x$ and $y$, and the large deviation functions of the marginal distributions of $x$, and of $y$. These marginal distributions match continuously with "near tails" of the $x$ and $y$ distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint $x$ and $y$ distribution $P(x,y,t)$ in a vicinity of a special "zero-noise" point, and show that $\ln P(x,y,t)$ has a nontrivial self-similar structure as a function of $x$, $y$ and $t$. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal $x$- and $y$-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times.

arXiv comment12 pages including 8 figures

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• Hard core run and tumble particles on a one dimensional lattice
• Nonequilibirum steady state for harmonically-confined active particles
• Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries
• Active Brownian motion with speed fluctuations in arbitrary dimensions: exact calculation of moments and dynamical crossovers
• Spectral density of individual trajectories of an active Brownian particle
• Most probable path of an active Brownian particle
• Self-propulsion with speed and orientation fluctuation: exact computation of moments and dynamical bistabilities in displacement
• Confined run and tumble particles with non-Markovian tumbling statistics
• Dynamical crossovers and correlations in a harmonic chain of active particles
• Tagged particle behavior in a harmonic chain of direction reversing active Brownian particles
• Active Brownian particle under stochastic position and orientation resetting in a harmonic trap
• Chirality Reversing Active Brownian Motion in Two Dimensions
• Short-time large deviation of constrained random acceleration process
• Telomeres in Lamin-A Depleted Cells Exhibit Directed Motion and Dynamic Coherence
• Irreversibility of stochastic state transitions in Langevin systems with odd elasticity
• Universal framework for the long-time position distribution of free active particles
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• From Attraction to Repulsion: Emergent Interactions in Harmonically Coupled Active Binary System