Effective diffusivity in active Brownian suspensions

We study the single-particle diffusion of a Brownian probe of size $R$ in a suspension comprised of a Newtonian solvent and a dilute dispersion of active Brownian particles (ABPs) with size $a$, characteristic swim velocity $U_0$, and a reorientation time $\tau_R$. These ABPs, or “swimmers,” have a run length $\ell = U_0 \tau_R$, and a mechanical activity $k_s T_s = \zeta_a U_0^2 \tau_R /6$, where $\zeta_a$ is the Stokes drag coefficient of a swimmer. When the swimmers are inactive, collisions between the probe and the swimmers sterically hinder the probe's diffusive motion. When the activity of the swimmers is greater than the Boltzmann energy, $k_s T_s > k_B T$, rather than being sterically hindered, the probe diffusivity is actually greater than its Stokes-Einstein-Sutherland diffusivity due to the mechanical energy imparted to the probe upon collisions with the swimmers. The active contribution to the effective diffusivity is a non-monotonic function of the swimmers' run length compared to the contact length between the probe and a swimmer: $\ell/(R+a)$. Comparisons are made to previous theoretical and experimental investigations of the hydrodynamic diffusion of a colloidal particle in a dilute suspension of swimming bacteria.