Random walk method

In physics, the random walk method has already been in use for decades to understand and model diffusion processes. Prickett et al. (1981) developed a simple model for groundwater transport to calculate the migration of contamination. An essential advantage of the methods of random walk and particle tracking is that they are free of numeric dispersion and oszillations (Abbot 1966). 

Eloot etal. suggested a new general matrix method for calculations involving noncylindrical pores, in which the pore is divided into sections and for each section a transmission line model with constant impedances is used. Direct simulations of the impedances for porous electrodes were also carried out using a random walk method. ... 

Relaxation times are commonly measured for porous media that have been saturated with a fluid such as water or an aqueous brine solution. The observed relaxation times are strongly dependent on the pore size, the distribution of pore sizes, the type of material (e.g. content of paramagnetic ions) and the water content. While relaxation times in porous media have been modelled using random walk methods and finite-element methods, simplified models are usually needed to obtain information on pore space. Section 3.2 reviews the standard model used to analyse relaxation behaviour of fluid in macroporous samples such as rocks. Mesoporous materials such as porous silica will be discussed in Section 3.3. 

To make it possible to deal with systems with many degrees of freedom, the Boltzmann operator/time evolution operators are represented by a Feynman path integraland the path integral evaluated by a Monte Carlo random walk method. It is in general not feasible to do this for real values of the time t, however, because the integrand of the path integral would be oscillatory. We thus first calculate for real values of t it, i.e., pure imaginary time,... 

Calculations have been carried out by a statistical method [23,a] and also by a random-walk method [23,d] (Section 2.5.1.2). They indicate that, for values of ab less than 0.1, the ratio (k/ko)/0A0B would be in the region of 3-10. According to the latter treatment, for example, the factor (for equal-sized molecules) is [1 - - (2r g/3)( ) g/Z)AB)] which with Equations (2.4) and (2.26) becomes [1 - - (8/3)(n/m)]. With the Stokes-Einstein values n = 6 and m = 8, the factor is therefore 3 with the more realistic values n = 3 — 4 and m = 3 — 4, it is in the region 4 1. The effect is appreciable it imphes, for instance, that reactions with 0a0b down to 0.1 would be difficult to differentiate experimentally from reactions with 0a0B 1 ... 

Although the number of jumps is tremendously large, the mean displacement of each atom is relatively small - most of the time it moves back and forth. In the diffusion process it is not possible to observe the individual jumps of the atoms, and it is necessary to find a relation between the individual atom jumps for large number of atoms and the diffusion phenomena which may be observed on a macroscopic scale. The problem is to find how far a large number of atoms will move from their original sites after having made a large number of jumps. Such relations may be derived statistically by means of the so-called random walk method. 

J. B. Anderson and B. H. Freihaut,/. Comput. Phys., 31,425 (1979). Quantum Chemistry by Random Walk Method of Successive Corrections. 

Polymer models were created using the Amorphous Cell module of the Materials Studio suite of software based on the self-avoiding random walk method of Theodorou and Suter and on the Meirovitch scanning method. Amorphous cis-PBD three-dimensional (3D) models consisted of 10 chains of 30-monomer oligomers and were equilibrated using a temperature cycle protocol under periodic boundary conditions (Figure 9.1). For a full description of the methodology used to build the equilibrated polymer models at various temperatures, please see Ref. [7]. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

A. Static Methods self-avoiding random walks... 

A. Static Methods Self-avoiding Random Walks... 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

MMT (32) is a 1- or 2-dimensional solute transport numerical groundwater model, to be driven off-line by a flow transport, such as VTT (Variable Thickness Transport). MMT employs the random-walk numerical method and was originally developed for radionuclide transport. The model accounts for advection, sorption and decay. 

An approach that does not suffer from such problems is the ABF method. This method is based on computing the mean force on and then removing this force in order to improve sampling. This leads to uniform sampling along . The dynamics of corresponds to a random walk with zero mean force. Only the fluctuating part of the instantaneous force on remains. This method is quite simple to implement and leads to a very small statistical error and excellent convergence. 

There are two other methods in which computers can be used to give information about defects in solids, often setting out from atomistic simulations or quantum mechanical foundations. Statistical methods, which can be applied to the generation of random walks, of relevance to diffusion of defects in solids or over surfaces, are well suited to a small computer. Similarly, the generation of patterns, such as the aggregation of atoms by diffusion, or superlattice arrays of defects, or defects formed by radiation damage, can be depicted visually, which leads to a better understanding of atomic processes. 

Ziegler-Natta catalysts for, 26 536-540 Random scission initiation, 23 372 Random thermal motion, in silicon-based semiconductors, 22 237-238 Random walk process, 26 1022 Raney nickel catalyst, 74 48 77 121 Raney-type catalysts, 25 195 Range of ambivalence, 76 700 Range quantities, methods for obtaining, 74 432... 

The MC method considers the configuration space of a model and generates a discrete-time random walk through configuration space following a master equation41,51... 

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