# Dynamics-dependent density distribution in active suspensions

## TL;DR

This paper demonstrates that the density distribution of active colloids with spatially varying speeds follows a specific inverse relation, verified experimentally using light-controlled bacteria, highlighting non-equilibrium behavior.

## Contribution

It provides the first quantitative verification of the theoretical relation between density and speed in active suspensions with spatially dependent velocities.

## Key findings

- Density distribution follows ρ(x) ∝ 1/v(x) in steady state.
- Experimental verification using light-controlled bacteria.
- Shows non-equilibrium behavior distinct from thermal systems.

## 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 $\rho(x)$ satisfies $\rho(x) = \rho(0)v(0)/v(x)$, where $x = 0$ is an arbitrary reference point. Such a dependence of steady-state $\rho(x)$ 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, $\rho(x)v(x) = \mathrm{constant}$, provided that steady state is reached.