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Register a new userSteady-State Levy Flights in a Confined Domain
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S. I. Denisov
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We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric L{e}vy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for L{e}vy flights is derived and solved analytically in the steady state. It is shown that L{e}vy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified.
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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.
Among Markovian processes, the hallmark of Levy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Levy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.
Let L(t) be a Levy flights process with a stability index alphain(0,2), and U be an external multi-well potential. A jump-diffusion Z satisfying a stochastic differential equation dZ(t)=-U(Z(t-))dt+sigma(t)dL(t) describes an evolution of a Levy particle of an `instant temperature sigma(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. sigma(t)approx t^{-theta} for some theta>0. We discover two different cooling regimes. If theta<1/alpha (slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as tto infty, which is concentrated at the potentials local minima. If theta>1/alpha (fast cooling) the Levy particle gets trapped in one of the potential wells.
Levy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the jump lengths---are drawn from an $alpha$-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Levy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Levy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Levy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index $alpha$ and the skewness (asymmetry) parameter $beta$. The other approach is based on the stochastic Langevin equation with $alpha$-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.
We investigate the first-passage dynamics of symmetric and asymmetric Levy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Levy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Levy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.
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