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.
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机译：自推进胶体构成一类重要的内在非平衡物质。通常，此类粒子会在短时间内弹道运动，但最终会改变其方向，并在长时间限制下显示随机游走行为。理论预测，如果非互动游泳者的速度在1D空间v（x）中发生空间变化，则其密度ρ（x）满足ρ（x）=ρ（0）v（0）/ v（x），其中x = 0是任意参考点。在热平衡中，禁止稳态ρ（x）对粒子动力学的这种依赖，这是最近研究证明如何用细菌“涂抹”的定性基础。在这里，我们通过构建在光照时以强度依赖的速度游泳的细菌并实现空间分辨的差分动态显微镜（sDDM）进行毫米长度范围的定量分析，从而定量验证这一预测。因此，应用空间光模式会创建一个速度曲线，如果达到稳态，我们沿着该曲线会发现ρ（x）v（ x ）=恒定。
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