No abstract

For a wide class of stochastic athermal systems, we derive Langevin-like equations driven by non-Gaussian noise, starting from master equations and developing a new asymptotic expansion. We found an explicit condition whereby the non-Gaussian properties of the athermal noise become dominant for tracer particles associated with both thermal and athermal environments. Furthermore, we derive an inverse formula to infer microscopic properties of the athermal bath from the statistics of the tracer particle. We apply our formulation to a granular motor under viscous friction and analytically obtain the angular velocity distribution function. Our theory demonstrates that the non-Gaussian Langevin equation is the minimal model of athermal systems.

Brownian motion is widely used as a paradigmatic model of diffusion in equilibrium media throughout the physical, chemical, and biological sciences. However, many real world systems, particularly biological ones, are intrinsically outof-equilibrium due to the energy-dissipating active processes underlying their mechanical and dynamical features [1]. The diffusion process followed by a passive tracer in prototypical active media such as suspensions of active colloids or swimming microorganisms [2] indeed differs significantly from Brownian motion, manifest in a greatly enhanced diffusion coefficient [3-10], non-Gaussian tails of the displacement statistics [6,9,10], and crossover phenomena [9, 10] from non-Gaussian to Gaussian scaling. While such characteristic features have been extensively observed in experiments, there is so far no comprehensive theory explaining how they emerge from the microscopic active dynamics. Here we present a theoretical framework of the enhanced tracer diffusion in an active medium from its microscopic dynamics by coarse-graining the hydrodynamic interactions between the tracer and the active particles as a stochastic process. The tracer is shown to follow a non-Markovian coloured Poisson process that accounts quantitatively for all empirical observations. The theory predicts in particular a long-lived Lévy flight regime [11] of the 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, thus suggesting that the optimal foraging strategy of swimming microorganisms might crucially depend on the density in order to exploit the Lévy flights of nutrients [12]. Our framework not only provides the first validation of the celebrated * kiyoshi@sk.tsukuba.ac.jp Lévy flight model [11] from a physical microscopic dynamics, but can also be applied to address important conceptual questions, such as the thermodynamics of active systems [13], and practical ones regarding, e.g., the interaction of swimming microorganisms with nutrients and other small particles like degraded plastic [14] and the design of artificial nanoscale machines [15].A passive tracer immersed in a fluid at equilibrium moves randomly due to its collisions with the surrounding fluid molecules. Understanding how the observed stochastic process followed by the tracer relates to the statistical mechanics of the surrounding fluid, as accomplished in the seminal works by Einstein, Smoluchowski, and Langevin [16], has provided deep insight into the connection between molecular transport and equilibrium thermodynamics , which has been widely exploited to describe soft matter and other complex physical systems [17]. However, when either artificial self-propelled colloids or biological swimming micro-organisms, such as bacteria like Escherichia coli or algae like Volvox and Chlamydomonas reinhardtii [2], are also suspended, the diffusion of the tracer changes dramatically due to the active stirring of the fluid exerted by the self-propelled pa...

We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper (Kanazawa et al. in Phys Rev Lett 114:090601-090606, 2015). We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.

We perform three-dimensional simulations of the impact of a granular jet for both frictional and frictionless grains. Small shear stress observed in the experiment [X. Cheng et al., Phys. Rev. Lett. 99, 188001 (2007)] is reproduced through our simulation. However, the fluid state after the impact is far from a perfect fluid, and thus the similarity between granular jets and quark gluon plasma is superficial because the observed viscosity is finite and its value is consistent with the prediction of the kinetic theory.