Long-time position distribution of an active Brownian particle in two dimensions

We study the late-time dynamics of a single active Brownian particle in two dimensions with speed v_{0} and rotation diffusion constant D_{R}. We show that at late times t≫D_{R}^{-1}, while the position probability distribution P(x,y,t) in the x-y plane approaches a Gaussian form near its peak describing the typical diffusive fluctuations, it has non-Gaussian tails describing atypical rare fluctuations when sqrt[x^{2}+y^{2}]∼v_{0}t. In this regime, the distribution admits a large deviation form, P(x,y,t)∼exp{-tD_{R}Φ[sqrt[x^{2}+y^{2}]/(v_{0}t)]}, where we compute the rate function Φ(z) analytically and also numerically using an importance sampling method. We show that the rate function Φ(z), encoding the rare fluctuations, still carries the trace of activity even at late times. Another way of detecting activity at late times is to subject the active particle to an external harmonic potential. In this case we show that the stationary distribution P_{stat}(x,y) depends explicitly on the activity parameter D_{R}^{-1} and undergoes a crossover, as D_{R} increases, from a ring shape in the strongly active limit (D_{R}→0) to a Gaussian shape in the strongly passive limit (D_{R}→∞).