Tracer diffusion in active suspensions

Abstract

We study the diffusion of a Brownian probe particle of size $R$ in a dilute dispersion of active Brownian particles (ABPs) of size $a$, characteristic swim speed $U_0$, reorientation time $τ_R$, and mechanical energy $k_s T_s = ζ_a U_0^2 τ_R /6$, where $ζ_a$ is the Stokes drag coefficient of a swimmer. The probe has a thermal diffusivity $D_P = k_B T/ζ_P$, where $k_B T$ is the thermal energy of the solvent and $ζ_P$ is the Stokes drag coefficient for the probe. When the swimmers are inactive, collisions between the probe and the swimmers sterically hinder the probe's diffusive motion. In competition with this steric hindrance is an enhancement driven by the activity of the swimmers. The strength of swimming relative to thermal diffusion is set by $Pe_s = U_0 a /D_P$. The active contribution to the diffusivity scales as $Pe_s^2$ for weak swimming and $Pe_s$ for strong swimming, but the transition between these two regimes is nonmonotonic. When fluctuations in the probe motion decay on the time scale $τ_R$, the active diffusivity scales as $k_s T_s /ζ_P$: the probe moves as if it were immersed in a solvent with energy $k_s T_s$ rather than $k_B T$.