Biased random walk

Averaging over all possible realizations of N steps, the average of i and is given by 

If the walk involves very large number of steps (N oo) with finite values of p (so that pN oo), PNini) of Equation 6.3 is approximated by the Gaussian distribution. 

This is derived by making the Stirling approximation for factorials and expanding PNini) around its maximum value at ( i and keeping only the quadratic term in the expansion. 

The above random walk process can be equivalently described as the probability that the walker is at x after N + 1 steps to be the sum of two terms p times the probability of being at a — 1 after N steps, and q times the probability of being at a -f 1 after N steps. This is stated as the master equation,  

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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. 

The power parameter q equals 2 for unbiased diffusive hopping from the donor to the acceptor, and between 1 and 2 for a direction-biased random walk process (Priyadarshy et al. 1996). 

When treating the above model numerically, we separate the motion of the vacancy and the tracer atom, as has been performed also in some of the analytical treatments referred to in Section 3.1. In our case of a finite lattice, this separation introduces an approximation, which is valid only if the tracer atom is relatively close to the middle of the lattice. First, we calculate the probabilities that the vacancy, released at one atomic spacing from the tracer, returns the first time to the tracer from equal (/ eq), perpendicular (/ perp) or opposite (popp) directions we also calculate the probability of its recombination (Prec) at the perimeter instead of returning to the tracer. Knowing these return and recombination probabilities, we calculate the statistics of the motion of the tracer atom, which performs a biased random walk of finite length. The probability distribution of the direction of each move with respect to the previous one, and the probability that a move was the last one, are obtained from the return and the recombination probabilities. 

Within Flory s mean field approximation, biased random walk statistics are characterized by ... 

Two different models of the biased random walk were envisaged. In model... 

Overall, the biased random walk, which places more emphasis on the motion toward the output end and less on the other directions, mimics more closely the transit profile of the experimental data. Both diffusion models, i.e., the blind and the myopic ant models, can reproduce the basic features of the real small-intestinal transit profile. 

Dunlap et al. assumed that transport can be described by a biased random walk through nearest-neighbor molecules having random but correlated Gaussian energies of zero mean and width a > AT. The analysis of Dunlap et al. leads to... 

The choice of a Gaussian distribution can be justified by the observation that the optical absorption profiles of well-defined conjugated moieties are Gaussian. Charge transport is treated as a biased random walk amongst the conjugated moieties, which have random site energies described by Equation... 

Figure 1. Biased random walk with a constant step size — Rosenbrock s fiinction. 

Together Eqs (7.18) and (7.23) express the essential features of biased random walk A drift with speed v associated with the bias kr ki, and a spread with a diffusion coefficient D. The linear dependence of the spread (fe ) on time is a characteristic feature of normal diffusion. Note that for a random walk in an isotropic three-dimensional space the corresponding relationship is... 

Transition path sampling is an importance sampling of trajectories, akin to the importance sampling of configurations described in Section II.E. Specifically, it is a biased random walk in the space of trajectories, in which each pathway is visited in proportion to its weight in the transition path ensemble. Because trajectories that do not exhibit the transition of interest have zero weight in this ensemble, they are never visited. In this way, attention is focused entirely on the rare but important trajectories, those that are reactive. 

Cells migrating in a chemoattractant gradient exhibit a biased random walk, with migration persisting in the direction of the gradient because they turn less and less often than during random migration. If a cell is... 

Allan, R.B. and Wilkinson, RC. (1978). A visual analysis of chemotactic and chem-kinetic locomotion of human neutrophil leucocytes. Exp. Cell Res. Ill, 191-203. Alt, W. (1980). Biased random walk models for chemotaxis and related diffusion approximations, y. Math. Biol. 9, 147-177. 

Reaction-Biased Random Walks. Propagation Failure 175... 

In this section we briefly derive a more general stochastic differential equation as the limit of a biased random walk. The treatment here is intuitive, based on the concepts of limiting processes and stochastic integration introduced above. For a rigorous treatment of the material presented here, it is recommended to consult a reference such as the excellent book of Nelson [279] (which also contains a very lively historical discussion). 

We shall derive, heuristically, a core piece of mathematical apparatus which permits much of the analysis of SDEs. Let 0(-) be any continuously twice differentiable function and let X e R represent the state of a biased random walk as considered in Sect. 6.3, i.e., with state X +i = x + a(X , n6i)8t + b(X , nSi) n x. Let 4>n = then Taylor s theorem gives... 

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