Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks (original) (raw)
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L\'evy walks in two dimensions: when microscopic symmetries cause macroscopic non-universality
arXiv (Cornell University), 2016
It is recognised now that a variety of real-life phenomena ranging from diffuson of cold atoms to motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.
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EPL (Europhysics Letters), 2014
Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of Lévy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such Lévy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.
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Physical Review E - PHYS REV E, 1998
We present a model of one-dimensional asymmetric random walks. Random walkers alternate between flights ͑steps of constant velocity͒ and sticking ͑pauses͒. The sticking time probability distribution function ͑PDF͒ decays as P(t)ϳt Ϫ . Previous work considered the case of a flight PDF decaying as P(t)ϳt Ϫ ͓Weeks et al., Physica D 97, 291 ͑1996͔͒; leftward and rightward flights occurred with differing probabilities and velocities. In addition to these asymmetries, the present strongly asymmetric model uses distinct flight PDFs for leftward and rightward flights: P L (t)ϳt Ϫ and P R (t)ϳt Ϫ , with . We calculate the dependence of the variance exponent ␥ ( 2 ϳt ␥ ) on the PDF exponents , , and . We find that ␥ is determined by the two smaller of the three PDF exponents, and in some cases by only the smallest. A PDF with decay exponent less than 3 has a divergent second moment, and thus is a Lévy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive (1Ͻ␥Ͻ2). When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive (␥Ͻ1); this is in contrast with the case with ϭ.
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Physical Review E, 2021
We introduce a persistent random walk model with finite velocity and self-reinforcing directionality, which explains how exponentially distributed runs self-organize into truncated Lévy walks observed in active intracellular transport by Chen et. al. [Nat. mat., 2015]. We derive the nonhomogeneous in space and time, hyperbolic PDE for the probability density function (PDF) of particle position. This PDF exhibits a bimodal density (aggregation phenomena) in the superdiffusive regime, which is not observed in classical linear hyperbolic and Lévy walk models. We find the exact solutions for the first and second moments and criteria for the transition to superdiffusion.
Emergence of Lévy walks in systems of interacting individuals
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We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)2041-172310.1038/ncomms9396] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the...
Lévy walk approach to anomalous diffusion
Physica A: Statistical Mechanics and its Applications, 1990
The transport properties of L6vy walks are discussed in the framework of continuous time random walks (CTRW) with coupled memories. This type of walks may lead to anomalous diffusion where the mean squared displacement (r2(t))~ t ~ with ct ~ 1. We focus on the enhanced diffusion limit, ct > 1, in one dimension and present our results on (r2(t)), the mean number of distinct sites visited S(t) and P(r, t), the probability of being at position r at time t.
Journal of Computational and Applied Mathematics, 2008
We consider a nearest neighbor, symmetric random walk on a homogeneous, ergodic random lattice Z d . The jump rates of the walk are independent, identically Bernoulli distributed random variables indexed by the bonds of the lattice. A standard result from the homogenization theory, see [A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick, An invariance principle for reversible Markov processes, Applications to random walks in random environments, J. Statist. Phys. 55(3/4) (1989) 787-855], asserts that the scaled trajectory of the particle satisfies the functional central limit theorem. The covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk. We use the duality structure corresponding to the product Bernoulli measure to construct a numerical scheme that approximates this parameter when d 3. The estimates of the convergence rates are also provided. © 2007 Published by Elsevier B.V. MSC: Primary 65C35; 82C41; Secondary 65Z05 Keywords: Eandom walk on a random lattice; Corrector; Duality ( ) t converge weakly, in P-probability with respect to , to the law of a Gaussian
From Random Walk to Single-File Diffusion
Physical Review Letters, 2005
We report an experimental study of diffusion in a quasi-one-dimensional (q1D) colloid suspension which behaves like a Tonks gas. The mean squared displacement as a function of time is described well with an ansatz encompassing a time regime that is both shorter and longer than the mean time between collisions. The ansatz asserts that the inverse mean squared displacement is the sum of the inverse mean squared displacement for short time normal diffusion (random walk) and the inverse mean squared displacement for asymptotic single-file diffusion (SFD). The dependence of the 1D mobility in the SFD on the concentration of the colloids agrees quantitatively with that derived for a hard rod model, which confirms for the first time the validity of the hard rod SFD theory. We also show that a recent SFD theory by Kollmann [Phys. Rev. Lett. 90, 180602 (2003)] leads to the hard rod SFD theory for a Tonks gas.
Taming Lévy flights in confined crowded geometries
The Journal of Chemical Physics, 2015
We study a two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is the same as in our previous work [J. Chem. Phys. 140, 044706 (2014)] in which standard, gaussian diffusion was studied. Here, a tracer is allowed to perform Cauchy random walk with uncorrelated steps. Our analysis shows that presence of obstacles significantly influences motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single trajectory realization and after the ensemble averaging. In particular, due to obstacles, the sample mean squared displacement asymptotically grows sublinearly in time, suggesting non-Markov character of motion. Closer inspection of survival probabilities indicates however that underlying diffusion is memoryless over long time scales despite strong inhomogeneity of motion induced by orientational ordering.