Random Walks on Lattices. II
01 Feb 1965-Journal of Mathematical Physics (American Institute of Physics)-Vol. 6, Iss: 2, pp 167-181
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.
Abstract: Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points.The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a1n + a2n½ + a3 + a4n−½ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited.The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c.Most of the results in this paper have been derived by the method of Green's functions.
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01 Jan 1979TL;DR: In this article, a brief description is given of the various manifestations of the universal fractional power law relaxation processes, which are contrasted with the classical or Debye law, and a novel very general approach based on the so-called energy criterion is introduced.
Abstract: A brief description is given of the various manifestations of the universal fractional power law relaxation processes, which are contrasted with the classical or Debye law. It is shown that the universal law is indeed found in a remarkable variety of physical and chemical situations, and this is deemed to merit a special attempt at finding a suitably general theoretical model. Several such models are briefly described, and a novel very general approach based on the so-called energy criterion is introduced. It is concluded that it is not yet possible to establish with certainty the validity of any of the models. >
4,012 citations
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.
Abstract: The website wheresgeorge.com invites its users to enter the serial numbers of their US dollar bills and track them across America and beyond. Why? “For fun and because it had not been done yet”, they say. But the dataset accumulated since December 1998 has provided the ideal raw material to test the mathematical laws underlying human travel, and that has important implications for the epidemiology of infectious diseases. Analysis of the trajectories of over half a million dollar bills shows that human dispersal is described by a ‘two-parameter continuous-time random walk’ model: our travel habits conform to a type of random proliferation known as ‘superdiffusion’. And with that much established, it should soon be possible to develop a new class of models to account for the spread of human disease. The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools1,2,3. Human travel, for example, is responsible for the geographical spread of human infectious disease4,5,6,7,8,9. In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic10, the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale-free random walks known as Levy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show 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 conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process.
2,120 citations
TL;DR: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes as mentioned in this paper, and a large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker-Planck equation.
Abstract: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.
2,119 citations
Cites methods from "Random Walks on Lattices. II"
...A convenient generalization of this process is the so-called continuous time random walk (CTRW) process, originally introduced by Montroll and Weiss (1965), in which both jump length and waiting time are distributed according to two PDFs, λ(x) and ψ(t)....
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TL;DR: Empirical data is used to show that the predictions of the CTRW models are in systematic conflict with the empirical results, and two principles that govern human trajectories are introduced, allowing for a statistically self-consistent microscopic model for individual human mobility.
Abstract: Individual human trajectories are characterized by fat-tailed distributions of jump sizes and waiting times, suggesting the relevance of continuous-time random-walk (CTRW) models for human mobility. However, human traces are barely random. Given the importance of human mobility, from epidemic modelling to traffic prediction and urban planning, we need quantitative models that can account for the statistical characteristics of individual human trajectories. Here we use empirical data on human mobility, captured by mobile-phone traces, to show that the predictions of the CTRW models are in systematic conflict with the empirical results. We introduce two principles that govern human trajectories, allowing us to build a statistically self-consistent microscopic model for individual human mobility. The model accounts for the empirically observed scaling laws, but also allows us to analytically predict most of the pertinent scaling exponents.
1,174 citations
TL;DR: In this article, a transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are Le´vy's a-stable densities, which represent plumesthat spread proportional to time 1/a, have heavy tails, and incorporate any degree of skewness.
Abstract: . A transport equation that uses fractional-order dispersion derivatives hasfundamental solutions that are Le´vy’s a-stable densities. These densities represent plumesthat spread proportional to time 1/a , have heavy tails, and incorporate any degree ofskewness. The equation is parsimonious since the dispersion parameter is not a functionof time or distance. The scaling behavior of plumes that undergo Le´vy motion isaccounted for by the fractional derivative. A laboratory tracer test is described by adispersion term of order 1.55, while the Cape Cod bromide plume is modeled by anequation of order 1.65 to 1.8. 1. Introduction Anomalous, or non-Fickian, dispersion has been an activearea of research in the physics community since the introduc-tion of continuous time random walks (CTRW) by Montrolland Weiss [1965]. These random walks extended the predictivecapability of models built on the stochastic process of Brown-ian motion, which is the basis for the classical advection-dispersion equation (ADE). The CTRW assign a joint space-time distribution, called the transition density, to individualparticle motions. When the tails are heavy enough (i.e., powerlaw), non-Fickian dispersion results for all time scales andspace scales.
1,106 citations
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Book•
01 Jan 1964
TL;DR: In this article, a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space, is studied, and the author considered this high degree of specialization worth while because of the theory of such random walks is far more complete than that of any larger class of Markov chains.
Abstract: This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of specialization worth while, because of the theory of such random walks is far more complete than that of any larger class of Markov chains. The book will present no technical difficulties to the readers with some solid experience in analysis in two or three of the following areas: probability theory, real variables and measure, analytic functions, Fourier analysis, differential and integral operators. There are almost 100 pages of examples and problems.
1,605 citations
562 citations