# Toward the full short-time statistics of an active Brownian particle on the plane – Archive ouverte HAL

### Satya N. Majumdar 1 Baruch Meerson

#### Satya N. Majumdar, Baruch Meerson. Toward the full short-time statistics of an active Brownian particle on the plane. Physical Review E , American Physical Society (APS), 2020, 102 (2), ⟨10.1103/PhysRevE.102.022113⟩. ⟨hal-03017046⟩

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.

• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques