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Register a new userTracer Dynamics in Crowded Active-Particle Suspensions
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We derive equations of motion for the mean-squared displacement (MSD) of an active Brownian particle (ABP) in a crowded environment modeled by a dense system of passive Brownian particles, and of a passive tracer particle in a dense active-Brownian particle system, using a projection-operator scheme. The interaction of the tracer particle with the dense host environment gives rise to strong memory effects. Evaluating these approximately in the framework of a recently developed mode-coupling theory for the glass transition in active Brownian particles (ABP-MCT), we discuss the various regimes of activity-induced super-diffusive motion and density-induced sub-diffusive motion. The predictions of the theory are shown to be in good agreement with results from an event-driven Brownian dynamics simulation scheme for the dynamics of two-dimensional active Brownian hard disks.
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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 behavior in the long-time limit. Theory predicts that if the velocity of non-interacting swimmers varies spatially in 1D, $v(x)$, then their density $rho(x)$ satisfies $rho(x) = rho(0)v(0)/v(x)$, where $x = 0$ is an arbitrary reference point. Such a dependence of steady-state $rho(x)$ on the particle dynamics, which was the qualitative basis of recent work demonstrating how to `paint with bacteria, is forbidden in thermal equilibrium. We verify this prediction quantitatively by constructing bacteria that swim with an intensity-dependent speed when illuminated. A spatial light pattern therefore creates a speed profile, along which we find that, indeed, $rho(x)v(x) = mathrm{constant}$, provided that steady state is reached.
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 out-of-equilibrium due to the energy-dissipating active processes underlying their mechanical and dynamical features. The diffusion process followed by a passive tracer in prototypical active media such as suspensions of active colloids or swimming microorganisms indeed differs significantly from Brownian motion, manifest in a greatly enhanced diffusion coefficient, non-Gaussian tails of the displacement statistics, and crossover phenomena 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 Levy flight regime 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 Levy flights of nutrients. Our framework provides the first validation of the celebrated Levy flight model from a physical microscopic dynamics.
We develop efficient numerical methods for performing many-body Brownian dynamics simulations of a recently-observed fingering instability in an active suspension of colloidal rollers sedimented above a wall [M. Driscoll, B. Delmotte, M. Youssef, S. Sacanna, A. Donev and P. Chaikin, Nature Physics, 2016, doi:10.1038/nphys3970]. We present a stochastic Adams-Bashforth integrator for the equations of Brownian dynamics, which has the same cost as but is more accurate than the widely-used Euler-Maruyama scheme, and uses a random finite difference to capture the stochastic drift proportional to the divergence of the configuration-dependent mobility matrix. We generate the Brownian increments using a Krylov method, and show that for particles confined to remain in the vicinity of a no-slip wall by gravity or active flows the number of iterations is independent of the number of particles. Our numerical experiments with active rollers show that the thermal fluctuations set the characteristic height of the colloids above the wall, both in the initial condition and the subsequent evolution dominated by active flows. The characteristic height in turn controls the timescale and wavelength for the development of the fingering instability.
We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels -- single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. We also present a survey of different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.
In a microrheological set-up a single probe particle immersed in a complex fluid is exposed to a strong external force driving the system out of equilibrium. Here, we elaborate analytically the time-dependent response of a probe particle in a dilute suspension of Brownian particles to a large step-force, exact in first order of the density of the bath particles. The time-dependent drift velocity approaches its stationary state value exponentially fast for arbitrarily small driving in striking contrast to the power-law prediction of linear response encoded in the long-time tails of the velocity autocorrelation function. We show that the stationary-state behavior depends nonanalytically on the driving force and connect this behavior to the persistent correlations in the equilibrium state. We argue that this relation holds generically. Furthermore, we elaborate that the fluctuations in the direction of the force display transient superdiffusive behavior.
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