Journal ArticleDOI
Random Walks with Infinite Spatial and Temporal Moments
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TLDR
The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory as mentioned in this paper, and the asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated.Abstract:
The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.read more
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The scaling laws of human travel
TL;DR: It is shown that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and concluded that human travel on geographical scales is an ambivalent and effectively superdiffusive process.
Journal ArticleDOI
Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.
TL;DR: This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion, and pays special attention to the ergodicity breaking parameters for the different anomalous stochastic processes.
Journal ArticleDOI
On the levy-walk nature of human mobility
TL;DR: A simple truncated Levy walk mobility (TLW) model is constructed that emulates the statistical features observed in the analysis and under which the performance of routing protocols in delay-tolerant networks (DTNs) and mobile ad hoc networks (MANETs) is measured.
Journal ArticleDOI
Fractional kinetic equations: solutions and applications
TL;DR: Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed, presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process.
Journal ArticleDOI
The fractional‐order governing equation of Lévy Motion
TL;DR: In this paper, a governing equation of stable random walks is developed in one dimension, which is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (a) of the highest derivative is fractional (e.g., the 1.65th derivative).
References
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Journal ArticleDOI
An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
Journal ArticleDOI
An Introduction to Probability Theory and Its Applications.
Journal ArticleDOI
Anomalous transit-time dispersion in amorphous solids
Harvey Scher,Elliott W. Montroll +1 more
TL;DR: In this paper, the authors developed a stochastic transport model for the transient photocurrent, which describes the dynamics of a carrier packet executing a time-dependent random walk in the presence of a field-dependent spatial bias and an absorbing barrier at the sample surface.
Journal ArticleDOI
Random Walks on Lattices. II
TL;DR: In this paper, the mean first passage times and their dispersion in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions have been derived by the method of Green's functions.
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