We study a polar tracer, having a concave surface, immersed in a two-dimensional suspension of active particles. Using Brownian dynamics simulations, we measure the distributions and auto-correlation functions of forces and torque exerted by active particles on the tracer. The tracer experiences a finite average force along its polar axis, while all the correlation functions show exponential decay in time. Using these insights we construct the full coarse-grained Langevin description for tracer position and orientation, where the active particles are subsumed into an effective self-propulsion force and exponentially correlated noise. The ensuing mesoscopic dynamics can be described in terms of five dimensionless parameters. We perform a thorough parameter study of the mean squared displacement, which illustrates how the different parameters influence the tracer dynamics, which crosses over from a ballistic to diffusive motion. We also demonstrate that the distribution of tracer displacements evolves from a non-Gaussian shape at early stages to a Gaussian behavior for sufficiently long times.

中文翻译：

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活性浴中极性示踪剂的有效朗之万方程

我们研究了一种极性示踪剂，它有一个凹面，浸入活性粒子的二维悬浮液中。使用布朗动力学模拟，我们测量了活性粒子施加在示踪剂上的力和扭矩的分布和自相关函数。示踪剂沿其极轴承受有限的平均力，而所有相关函数都显示出时间指数衰减。使用这些见解，我们为示踪剂的位置和方向构建了完整的粗粒度朗之万描述，其中活动粒子被包含在有效的自推进力和指数相关的噪声中。随后的细观动力学可以用五个无量纲参数来描述。我们对均方位移进行了彻底的参数研究，这说明了不同的参数如何影响示踪剂动力学，从弹道运动到扩散运动。我们还证明了示踪剂位移的分布从早期的非高斯形状演变为足够长时间的高斯行为。