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 U_0, reorientation time τ_R, and mechanical energy k_sT_s=ζ_aU^2_0τ_R/6, where ζ_a is the Stokes drag coefficient of a swimmer. The probe has a thermal diffusivity D_P=k_BT/ζ_P, where k_BT 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_0a/D_P. The active contribution to the diffusivity scales as Pe^2_s 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_sT_s/ζ_P: the probe moves as if it were immersed in a solvent with energy k_sT_s rather than k_BT.

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