Dynamics-dependent density distribution in active suspensions

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 behaviour 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. Here we verify this prediction quantitatively by constructing bacteria that swim with an intensity-dependent speed when illuminated and implementing spatially-resolved differential dynamic microscopy (sDDM) for quantitative analysis over millimeter length scales. Applying a spatial light pattern therefore creates a speed profile, along which we find that, indeed, ρ(x)v(x) = constant, provided that steady state is reached.