Diffusion coefficients for multi-step persistent random walks on lattices (original) (raw)

Random and deterministic walks on lattices

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

On multiple visits in lattice random walks

Zeitschrift f�r Physik B Condensed Matter, 1988

In the present work we treat in detail the problem of multiple visits in lattice random walks. We show that this problem is closely related to the well-studied property of the number of distinct sites visited at least one in an n-step walk. With simple algebraic manipulations we provide new analytical solutions for the mean number of sites visited exactly a certain number of times, and the mean number of sites visited at least a certain number of times. We find that the moments of these quantities vary asymptotically with time. The resulting exponents exhibit constant gap scaling. Computer simulations are given that are within excellent agreement with the derived expressions.

Exact results and self-averaging properties for random-random walks on a one-dimensional infinite lattice

Journal of Statistical Physics, 1989

We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocity V and the diffusion constant D (which are found to coincide with those given by Derrida) and for demonstrating that V is indeed a self-averaging quantity; the same property is established for D in the limiting case of a directed walk.

Random walks on periodic and random lattices. II. Random walk properties via generating function techniques

Journal of Statistical Physics, 1979

We investigate the random walk properties of a class of two-dimensional lattices with two different types of columns and discuss the dependence of the properties on the densities and detailed arrangements of the columns. We show that the row and column components of the mean square displacement are asymptotically independent of the details of the arrangement of columns. We reach the same conclusion for some other random walk properties (return to the origin and number of distinct sites visited) for various periodic arrangements of a given relative density of the two types of columns. We also derive exact asymptotic results for the occupation probabilities of the two types of distinct sites on our lattices which validate the basic conjecture on bond and step ratios made in the preceding paper in this series.

Random Walks on Randomly Oriented Lattices

Markov Processes and Related Fields

Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the orientation the asymptotic behaviour of the random walk is shown to be recurrent or transient. In particular, for randomly horizontally oriented lattices the random walk is almost surely transient.

Diffusion on two-dimensional random walks

Physical Review Letters, 1987

Analysis of Monte Carlo enumerations for diAusion on the fractal structure generated by the random walk on a two-dimensional lattice allows us to predict a behavior &r)n "(1nn)' with v=0. 325~0.01 and a =0.35~0.03. This leads to the conjecture that v=a = -, ' . This value of v, and the presence of logarithmic corrections, are strongly supported by heuristic arguments based on Flory theory and on plausible assumptions.

Memory effects in diffusions in a 2D fluctuating lattice

Physics Letters A, 1991

We consider a 2D random lattice: each node can be in two states, 0, 1, and fluctuates according to a Poisson process. A test particle is moving on the lattice and is scattered at each site depending on the state of the site. We investigate the transmission probabilities and the diffusion coefficient numerically, and we show that the diffusion remains non-Gaussian long after the diffusive regime is established. 158 0375-9601/9 17$ 03.50

Invariance relations for random walks on hexagonal lattices

Chemical Physics Letters, 2003

We consider the problem of random walks on finite, N ¼ ð2k  2kÞ hexagonal lattices with a single, deep trap, and subject to periodic boundary conditions. An exact expression is obtained for calculating the invariance relation linking the set M of nth nearest-neighbor sites surrounding the trapping site, viz., ð2M À 3ÞN À f2M þ 6 þ 3M½lnðM=6Þ= lnð2Þg: This result may be used to obtain approximate values of the overall mean walklength hni. The results are compared with exact numerical results, with the predictions of the asymptotic expression of Montroll and Weiss, and linked to current studies in nanotube chemistry.

The persistence length of two-dimensional self-avoiding random walks

Journal of Physics A: Mathematical and General, 2003

The decay of directional correlations in self-avoiding random walks on the square lattice is investigated. Analysis of exact enumerations and Monte Carlo data suggest that the correlation between the directions of the first step and the jth step of the walk decays faster than j −1 , indicating that the persistence length of the walk is finite.

On the temporal order of first-passage times in one-dimensional lattice random walks

Journal of Computational and Applied Mathematics, 2005

A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing modified Bessel functions of the first kind. By using several transformations simpler integrals are obtained from which for two and three particles asymptotic approximations are derived for large values of the parameters. Expressions of the probability for n particles are also derived. I returned and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.