Dynamics-dependent density distribution in active suspensions

Abstract

Self-propelled colloids constitute an important class of intrinsically non-equilibrium matter. Typically, such a particle moves ballistically at short times, but eventually changes its orientation, and displays random-walk behavior in the long-time limit. Theory predicts that if the velocity of non-interacting swimmers varies spatially in 1D, $v(x)$, then their density $ρ(x)$ satisfies $ρ(x) = ρ(0)v(0)/v(x)$, where $x = 0$ is an arbitrary reference point. Such a dependence of steady-state $ρ(x)$ on the particle dynamics, which was the qualitative basis of recent work demonstrating how to `paint' with bacteria, is forbidden in thermal equilibrium. We verify this prediction quantitatively by constructing bacteria that swim with an intensity-dependent speed when illuminated. A spatial light pattern therefore creates a speed profile, along which we find that, indeed, $ρ(x)v(x) = \mathrm{constant}$, provided that steady state is reached.