Brownian motion is widely used as a model of diffusion in equilibrium media throughout the physical, chemical and biological sciences. However, many real-world systems are intrinsically out of equilibrium owing to energy-dissipating active processes underlying their mechanical and dynamical features¹. The diffusion process followed by a passive tracer in prototypical active media, such as suspensions of active colloids or swimming microorganisms², differs considerably from Brownian motion, as revealed by a greatly enhanced diffusion coefficient^{3,4,5,6,7,8,9,10} and non-Gaussian statistics of the tracer displacements^6,9,10. Although these characteristic features have been extensively observed experimentally, there is so far no comprehensive theory explaining how they emerge from the microscopic dynamics of the system. Here we develop a theoretical framework to model the hydrodynamic interactions between the tracer and the active swimmers, which shows that the tracer follows a non-Markovian coloured Poisson process that accounts for all empirical observations. The theory predicts a long-lived Lévy flight regime¹¹ of the loopy tracer motion with a non-monotonic crossover between two different power-law exponents. The duration of this regime can be tuned by the swimmer density, suggesting that the optimal foraging strategy of swimming microorganisms might depend crucially on their density in order to exploit the Lévy flights of nutrients¹². Our framework can be applied to address important theoretical questions, such as the thermodynamics of active systems¹³, and practical ones, such as the interaction of swimming microorganisms with nutrients and other small particles¹⁴ (for example, degraded plastic) and the design of artificial nanoscale machines¹⁵.

布朗运动在整个物理、化学和生物科学中被广泛用作平衡介质中的扩散模型。^{然而，许多现实世界的系统由于其机械和动力学特征1}的能量耗散活动过程而本质上是不平衡的。扩散过程后跟在原型活性介质中的被动示踪剂，例如活性胶体或游泳微生物^{2的悬浮液，与布朗运动有很大不同，正如扩散系数3、4、5、6、7、8}大大增强所揭示的那样， ^9,10和示踪剂位移的非高斯统计^6,9,10. 尽管这些特征已经在实验中得到了广泛的观察，但迄今为止还没有全面的理论来解释它们是如何从系统的微观动力学中出现的。在这里，我们开发了一个理论框架来模拟示踪剂和活跃游泳者之间的流体动力学相互作用，这表明示踪剂遵循非马尔可夫彩色泊松过程，该过程解释了所有经验观察。该理论预测了长期的 Lévy 飞行状态¹¹在两个不同的幂律指数之间具有非单调交叉的环形示踪运动。这一制度的持续时间可以通过游泳者的密度进行调整，这表明游泳微生物的最佳觅食策略可能主要取决于它们的密度，以利用 Lévy 飞行的营养物质¹²。我们的框架可用于解决重要的理论问题，例如活性系统的热力学¹³和实际问题，例如游泳微生物与营养物质和其他小颗粒¹⁴（例如，降解塑料）的相互作用以及人工系统的设计纳米级机器¹⁵．