A lattice-model representation of continuous-time random walks

2010

This paper addresses the problem of the mesoscopic description of reaction-transport system of particles performing a continuous time random walk CTRW 1. One of the main challenges is an implementation of the description of chemical reactions in non-Markovian transport processes governed by CTRW with nonexponential waiting time distributions. There exist several approaches and techniques to deal with this problem 2–11.

Journal of Statistical Mechanics: Theory and Experiment, 2015

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.

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

2017

Diffusion-coagulation can be simply described by a dynamic where particles perform a random walk on a lattice and coalesce with probability unity when meeting on the same site. Such processes display non-equilibrium properties with strong fluctuations in low dimensions. In this work we study this problem on the fully-connected lattice, an infinite-dimensional system in the thermodynamic limit, for which mean-field behaviour is expected. Exact expressions for the particle density distribution at a given time and survival time distribution for a given number of particles are obtained. In particular we show that the time needed to reach a finite number of surviving particles (vanishing density in the scaling limit) displays strong fluctuations and extreme value statistics, characterized by a universal class of non-Gaussian distributions with singular behaviour.

Mathematical Modelling of Natural Phenomena, 2013

One of the central results in Einstein's theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

Scientific Computing, Validated Numerics, Interval Methods, 2001

Random walk methods are suitable to build up convergent solutions for reaction-diffusion problems and were successfully applied to simulations of transport processes in a random environment. The disadvantage is that, for realistic cases, these methods become time and memory expensive. To increase the computation speed and to reduce the required memory, we derived a "global random walk" method in which the particles at a given site of the grid are simultaneously scattered following the binomial Bernoulli repartition. It was found that the computation time is reduced three orders of magnitude with respect to individual random walk methods. Moreover, by suitable "microscopic balance" boundary conditions, we obtained good simulations of transport in unbounded domains, using normal size grids. The global random walk improves the statistical quality of simulations for diffusion processes in random fields. The method was tested by comparisons with analytical and finite difference solutions as well as with concentrations measured in "column experiments", used in laboratory study of soils' hydrogeological and chemical properties.

Computing and Visualization in Science, 2007

The Global Random Walk algorithm performs simultaneously the tracking of large collections of particles and permits massive simulations at reasonable costs. Applications were developed for transport in systems with anisotropic, non-homogeneous, and randomly distributed parameters. As a first illustration we present simulations for diffusion in human skin. Further, a case study for contaminant transport in groundwater shows that the realizations of the transport process converge in mean square limit to a Gaussian diffusion. This investigation also indicates that the use of the Kraichnan routine, based on periodic random fields, yields reliable simulations of transport in Gaussian velocity fields.

Stochastic Processes and Their Applications, 1992

Random walks of particles on a lattice are a classical paradigm for the microscopic mechanism underlying diffusive processes. In deterministic walks, the role of space and time can be reversed, and the microscopic dynamics can produce quite different types of behavior such as directed propagation and organization, which appears to be generic behaviors encountered in an important class of systems. The various aspects of classical and not so classical walks on latices are reviewed with emphasis on the physical phenomena that can be treated through a lattice dynamics approach. Comment: 13 pages including 3 figures; to appear in AIP Statistical Physics

Journal of Physics A: Mathematical and Theoretical

Diffusion-coagulation can be simply described by a dynamic where particles perform a random walk on a lattice and coalesce with probability unity when meeting on the same site. Such processes display non-equilibrium properties with strong fluctuations in low dimensions. In this work we study this problem on the fully-connected lattice, an infinite-dimensional system in the thermodynamic limit, for which mean-field behaviour is expected. Exact expressions for the particle density distribution at a given time and survival time distribution for a given number of particles are obtained. In particular we show that the time needed to reach a finite number of surviving particles (vanishing density in the scaling limit) displays strong fluctuations and extreme value statistics, characterized by a universal class of non-Gaussian distributions with singular behaviour.