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 of size a, characteristic swim speed U0, reorientation time τR, and mechanical energy ksTs=ζaU02τR/6, where ζa is the Stokes drag coefficient of a swimmer. The probe has a thermal diffusivity DP=kBT/ζP, where kBT is the thermal energy of the solvent and ζP 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 Pes=U0a/DP. The active contribution to the diffusivity scales as Pes2 for weak swimming and Pes for strong swimming, but the transition between these two regimes is nonmonotonic. When fluctuations in the probe motion decay on the time scale τR, the active diffusivity scales as ksTs/ζP: the probe moves as if it were immersed in a solvent with energy ksTs rather than kBT.