Self-diffusive dynamics of active Brownian particles at moderate densities

Abstract

The Active Brownian Particle (ABP) model has become a prototype of self-propelled particles. ABPs move persistently at a constant speed $V$ along a direction that changes slowly by rotational diffusion, characterized by a coefficient $\Dr$. Persistent motion plus random reorientations generate a random walk at long times with a diffusion coefficient that, for isolated ABPs in two dimensions, is given by $D_0=V^2/(2\Dr)$. Here we study the density effects on the self-diffusive dynamics using a recently proposed kinetic theory for ABPs, in which persistent collisions are described as producing a net displacement on the particles. On intermediate timescales, where many collisions have taken place but the director of the tracer particle has not yet changed, it is possible to solve the Lorentz kinetic equation for a tracer particle. It turns out that, as a result of collisions, the tracer follows an effective stochastic dynamics, characterized by an effective reduced streaming velocity $V_\text{eff}$ and anisotropic diffusion, with coefficients explicitly depending on the density. Based on this result, an effective theoretical and numerical approach is proposed in which the particles in a bath follow stochastic dynamics with mean-field interactions based on the local density. Finally, on time scales larger than $\Dr^{-1}$, studying the van Hove function at small wavevectors, it is shown that the tracer particle presents an effective diffusive motion with a coefficient $D=V_\text{eff}^2/(2\Dr)$. The dependence of $V_\text{eff}$ on the density indicates that the kinetic theory is limited to area fractions smaller than 0.42, and beyond this limit unphysical results appear.