Ordinary Random Walk Research Articles

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one “effectively independent” sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents areυ=0.7496±0.0007 ind=2 andυ= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200⩽N⩽10000 and 200⩽N⩽ 3000, respectively.

The winding angle distribution of an ordinary random walk

The winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers. The distribution of winding angles is studied here for ordinary or non-self-avoiding walks. New results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution. The results of this analysis are compared with simulations of random walks on a lattice, the agreement between the two is very good. There are, however, small and as yet unexplained discrepancies.

A unifying model of generalised random walks

The authors propose a unifying model for generalised static random walks, in which each walk has a weight exp(-g Sigma (ni)alpha ), where ni is the occupation number of a site and alpha and g are variable parameters. Special cases of this general walk include the ordinary (Brownian) random walk, the self-avoiding walk, the lattice Domb-Joyce model (1972) and the interacting random walk model recently introduced by Stanley et al. (1983). The asymptotic properties of the walk are studied in one dimension by effective medium arguments and exact enumeration methods. For repulsive correlations the authors find SAW behaviour, while for attractive correlations the model is self trapping for 1< alpha <or=2 and exhibits anomalous diffusion, with continuously varying exponents, for 0<or= alpha <1. In higher dimensions comments are made about effective medium predictions, and their relationship to other generalised random walk models.

Problems in the Use of Historical Data in Estimating Economic Loss in Wrongful Death and Injury Cases

In his Comment, William Harris suggests an alternative to first differencing as a means to achieve stationarity when dealing with nominal time series on interest rates and earnings.' Specifically, he suggests that time series expressed in real terms may be stationary and eliminate the problems associated with using nonstationary series. The use of reliable data expressed in real terms also would eliminate much of the controversy surrounding the proper treatment of expected future inflation in accidental death and injury awards. Unfortunately, the issue is not as simple as Harris makes it appear. There are a number of problems with his analysis. First, his conclusion that nominal data can be used to calculate the present value of future earnings requires a stronger condition that stationarity in the real data. Second, the problems involved in estimating real data were not addressed. Third, his interpretation of his empirical results is not accurate. At the outset we should be clear about what constitutes stationarity in time series data. Throughout his Comment, Harris misinterprets stationarity as implying no trend or that a series is . . relatively stable . . . (constant?). However, that is too restrictive. A series is stationary if the stochastic process that generates the series is stable or invariant with respect to time. The values assumed by the generated variable can vary with respect to time; a nonstationary series may or may not exhibit a trend. For example, the ordinary random walk is a trendless nonstationary process. Moreover, a trendless stationary series need not be a constant (with only random error as in a white noise process), but may exhibit a cyclical structure.

Statistics of one-dimensional cluster motion

The statistics of clusters, made up of metal atoms in adjacent one-dimensional diffusion channels, are developed quantitatively. Kolmogorov’s equation is used to find the mean square displacement for clusters capable of existing in energetically different configurations at the same position of the center of mass; this is done under steady state conditions, for which the probability of finding a specified configuration does not vary in time. Two systems are examined: (1) dimers capable of existing in an infinite number of states, a situation realized if dissociation is allowed, and (2) trimers diffusing on planes, such as W(211), on which nine distinct jump processes may contribute to the cluster motion. In dimer diffusion it is demonstrated that dissociation may be important even if the fraction dissociated is minor. For trimers, previous attempts to approximate the motion through the use of average transition rates are compared with the exact solutions and found wanting. Important statistical quantities beyond the mean square displacement are presented for simple dimers capable of existing in only two states. The generating function is derived, together with the higher moments of the displacement. Probability density functions for the number of jumps in an interval t, for the waiting time up to a specified jump, as well as for the displacements are all presented. These differ significantly from the density functions for an ordinary random walk. However, an averaging technique allows simple approximations for the behavior of dimers.

The oscillating random walk

{ Y n ; n=0, 1, …} denotes a stationary Markov chain taking values in R d . As long as the process stays on the same side of a fixed hyperplane E 0, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′ n = | Y′ n−1 − Z n | (with the Z n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.

Cation Diffusion and Conductivity in Solid Electrolytes. I

The characteristics of cation diffusion and ionic conductivity in solid electrolytes are discussed using β- and β″-alumina as examples. The problem is characterized by a cation migration by the vacancy mechanism in a “cation-disordered phase,” a system in which the number of available vacant sites is of the same order of magnitude as the number of diffusing cations. The path probability method is used to derive both the tracer diffusion and the ionic conductivity; this method avoids the difficulties connected with the application of the ordinary random walk approach to such systems. β″-Alumina is regarded as an example of the “cation-disordered phases” in which all available cation sites are equivalent, while β-alumina represents those in which available cation sites are divided into several nonequivalent sublattice sites. When the derived diffusion coefficient is interpreted using the language of the conventional random walk approach, the jump frequency of cations is found to include two factors, the vacancy availability factor V and the effective jump frequency factor W, both of which take into account the effect of surroundings of the migrating ion and characterize the cooperative mode of the cation motion. Both of these factors, as well as the correlation factor f, depend strongly on composition of cations and on temperature. The Nernst–Einstein relation which relates the tracer diffusion and the charge diffusion is found to include characteristic correlation factors. Even in ionic conductivity, a correlation factor fI is found to appear in the β-alumina-type electrolytes. Although both f and fI vary drastically depending on the cation concentration, the ratio of the two remains within limits determined by the geometry of the crystal lattice. The value of the ratio indicates the predominant mechanism of diffusion.