Random walks scaling dynamics

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. 

Brownian Dynamics (BD) methods treat the short-term behavior of particles influenced by Brownian motion stochastically. The requirement must be met that time scales in these simulations are sufficiently long so that the random walk approximation is valid. Simultaneously, time steps must be sufficiently small such that external force fields can be considered constant (e.g., hydrodynamic forces and interfacial forces). Due to the inclusion of random elements, BD methods are not reversible as are the MD methods (i.e., a reverse trajectory will not, in general, be the same as the forward using BD methods). BD methods typically proceed by discretization and integration of the equation for motion in the Langevin form... 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

The continuous limit of a simple random walk model leads to a stochastic dynamic equation, first discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. 

Note, however, that the conclusions drawn are valid only in the asymptotic long-time limit (t oo). At short time scales, the particle can be envisioned to execute a random walk in a potential field wherein the term corresponding to dominates the dynamics of the fluctuations of the potential field. In such a case, one can repeat the above analysis by utilizing the following functional form for the correlations of ... 

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