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Loopy Lévy flights enhance tracer diffusion in active suspensions


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 coefficient3,4,5,6,7,8,9,10 and non-Gaussian statistics of the tracer displacements6,9,10. 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.

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Fig. 1: Illustration of the microscopic model.
Fig. 2: Simulation results confirming the theoretical predictions.
Fig. 3: Force exerted on the tracer by the swimmers.

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Data and code availability

The data supporting the findings of this study are reproducible by a simulation code available at The time series data used in Figs. 1b, 3a are accessible at Other data are available from T.G.S. ( upon request.


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We thank D. Mizuno, H. Takayasu, M. Takayasu, H. Hayakawa and F. van Wijland for discussions. This work was supported by a Grant-in-Aid for JSPS Fellows (grant number 16J05315), JSPS KAKENHI (grant numbers 16K16016 and 18K13519) and a Research Fellowship granted by the Royal Commission for the Exhibition of 1851. A.C. acknowledges the hospitality of the Yukawa Institute for Theoretical Physics, Kyoto University as well as financial support. Simulations and numerical calculations were carried out using the Cray XC40 supercomputer at Yukawa Institute for Theoretical Physics in Kyoto University.

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Authors and Affiliations



K.K., T.G.S., A.C. and A.B. designed the research and interpreted the results. K.K. and A.C. performed analytical calculations. T.G.S. performed numerical simulations. A.B. managed the project. K.K., A.C. and A.B. wrote the paper.

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Correspondence to Kiyoshi Kanazawa.

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Peer review information Nature thanks Andreas Menzel and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

This file contains details on the simulation protocol and the theoretical framework including all technical calculations.

Video 1

Example trajectory of the passive tracer. This video shows a typical trajectory of the passive tracer and the corresponding force time series. The dynamics of the tracer is simulated according to equations (1) and (2).

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Kanazawa, K., Sano, T.G., Cairoli, A. et al. Loopy Lévy flights enhance tracer diffusion in active suspensions. Nature 579, 364–367 (2020).

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