Universal framework for the long-time position distribution of free active particles

Abstract

Active particles self-propel themselves with a stochastically evolving velocity, generating a persistent motion leading to a non-diffusive behavior of the position distribution. Nevertheless, an effective diffusive behavior emerges at times much larger than the persistence time. Here we develop a general framework for studying the long-time behaviour for a class of active particle dynamics and illustrate it using the examples of run-and-tumble particle, active Ornstein-Uhlenbeck particle, active Brownian particle, and direction reversing active Brownian particle. Treating the ratio of the persistence-time to the observation time as the small parameter, we show that the position distribution generically satisfies the diffusion equation at the leading order. We further show that the sub-leading contributions, at each order, satisfies an inhomogeneous diffusion equation, where the source term depends on the previous order solutions. We explicitly obtain a few sub-leading contributions to the Gaussian position distribution. As a part of our framework, we also prescribe a way to find the position moments recursively and compute the first few explicitly for each model.

Dynamical fluctuations of a tracer coupled to active and passive particles

“…The exponentially decaying autocorrelation is a hallmark of active particle motion [62,63]. Popular models like run-and-tumble particles [64][65][66] and active Ornstein-Uhlenbeck particles [67] exhibit exponentially decaying correlation of the propulsion velocity at all times.…”

Long time behavior of run-and-tumble particles in two dimensions

“…In the following, we solve the above equation perturbatively, by treating ε 2 /t as a small parameter following the framework developed recently [43]. To explicitly obtain the coefficient of the (ε 2 /t) k term in the perturbative series, the knowledge of the position moments ⟨y 2k (t)⟩ is required.…”

“…Proceeding similarly, one can systematically calculate the higher order corrections. Finally, remembering that the isotropy of two-dimensional position distribution, the radial distribution of the RTP in the diffusive scaling limit can be expressed in the universal form [43],…”

“…In fact, there is no general formalism to understand these active dynamics, and one has to approach the different models of self-propulsion in different ways to extract their statistical properties. Recently, however, it was shown [43] that the long-time dynamics of active particles show certain universal behavior irrespective of the specific dynamics of the stochastic propulsion velocity, namely, the leading order position distribution is a Gaussian with a diffusive scaling and the subleading corrections to this leading order Gaussian follow an inhomogeneous diffusion equation. The specificity of the different active dynamics enters through the source term only, which, at each order, depends on the previous order solutions.…”

“…To understand the effects of the activity of an RTP at late times via the large deviation functions experimentally is challenging as the O(t) events are rare. In this paper, we study the O( √ t) fluctuations beyond Gaussian, of the two-dimensional RTP at long-times using the formalism developed in [43]. Starting from the Fokker-Planck equation, we show that at the leading order, the position distribution satisfies a diffusion equation, which yields a leading order Gaussian position distribution.…”