Nearest neighbor random walks

When aR > 1 and aL S> 1, we have p(il) p( l) for i > 1, and the walk becomes a nearest neighbor random walk. The mean displacement and mean square displacement are then given by... 

The corresponding equations for the one-dimensional, discrete, nearest neighbor random walk are obtained by setting a — °°. They are... 

At a very low temperature where an adatom jumps only occasionally, about one atomic jump in every few seconds, field ion microscope studies conclude that the surface diffusion of adatoms is consistent with a discrete nearest neighbor random walk. However, in molecular dynamic simulations of diffusion phenomena, which are carried out only for high temperature diffusions where atomic jumps are very rapid, i.e. an atomic... 

In the random walk description, the energy transfer rate constant is related to the number of hops made by the exciton without revisiting any site in its lifetime. The nearest-neighbor random walk in an isotropic medium may be approximated by a random walk on a simple cubic lattice [145]... 

In Chapter 7 we saw examples of evolution equations in probability space that were constructed via the phenomenological route. Equation (7.3) for the nearest neighbor random walk problem,... 

In this section are displayed graphically the numerically exact results that have been obtained for unbiased, nearest-neighbor random walks on finite d = 2,3 dimensional regular. Euclidean lattices, each of uniform valency v, subject to periodic boundary conditions, and with a single deep trap. These data allow a quantitative assessment of the relative importance of changes in system size N, lattice dimensionality d, and/or valency v on the efficiency of diffusion-reaction processes on lattices of integral dimension, and provide a basis for understanding processes on lattices of fractal dimension or fractional valency. 

Table III.5 gives a comparison of these two representations of the data for (n) for nearest-neighbor random walks on finite, cubic lattices with a centrosymmetric trap and subject to periodic boundary conditions. A similar analysis [14] shows that for d = 3, tetrahedral lattices (v — 4)...

Recall that in the studies of Montroll and Weiss [17-19] on nearest-neighbor random walks on an infinite, periodic lattice of unit cells, the mean walklength ( ) is completely determined once the dimensionality d, the system size (number of lattice sites) N, and the connectivity (or valency) v of the unit cell are specified. For the class of d — 2 problems considered here, there is, not unexpectedly, a more subtle dependence of n) on the lattice... 

It is in the dependence of (n) on the (average) valency (u) that the results here stand in contrast to the analytic and numerical results obtained for lattices subject to periodic boundary conditions. From studies on periodic lattices, n) should decrease systematically with increase in the uniform valency v. This result pertains as well to random walks on ci = 3 dimensional periodic lattices of unit cells and can also be demonstrated analytically and numerically for walks on higher-dimensional [d < 8) cubic lattices [15,16]. In these problems, v = 2d and hence the higher the dimensionality of the space, the greater the number of pathways to a centrally located deep trap in a periodic array of (cubic) cells the decrease in (n) is found to be quite dramatic with increase in d, and hence v. However, an increase in v will also result in a greater number of pathways that allow the random walker to move away from the trap. For periodic lattices, this latter option positions the random walker closer to the trap in an adjacent cell. For finite lattices, moving away from the trap does not position the walker closer to a trap in an adjacent unit cell it positions the walker closer to the finite boundary of the lattice from whence it must (eventually) work itself back. It is evident, therefore, why the v dependence for periodic lattices is modified when one studies the same class of nearest-neighbor random-walk problems on finite lattices. 

Finally, as noted earlier, ratio n) v = 6)/ n) v = 4) equals 3/2 exactly for the case of the unbiased, nearest-neighbor random walk. As is seen in Table 1V.4, for attractive Coulombic interactions there is a systematic convergence to the exact limiting value of 3/2 with an increase in system size. Interestingly, the value 1.500 is realized for repulsive Coulombic interactions, even for the smallest system size reported. 

From the classical literature on continuum theories of diffusion-reaction processes based on Eq. (4.1), it is anticipated that the larger the system size, the longer the time scale required for the reactive event, Eq. (4.2), to occur. The corresponding dependence for lattice systems was first proved analytically by Montroll and Weiss [17-19] who studied nearest-neighbor random walks on finite lattices of integral dimension subject to periodic boundary conditions. In a lattice-based approach to diffusion-controlled processes, one can also examine the influence of the number of pathways (or reaction channels) available to the diffusing coreactant at each point in the... 

FIGURE 6.6 Mutual information for ethane, propane, and ethanethiol. Error bars mark averages 1 standard deviation. Quantities are based on the statistical structure of the formula diagrams subject to a nearest-neighbor random walk. 

Note that (8.72) implies that Xlm Kmn = 0 for all n. This is compatible with the fact that i) = 1 is independent of time. The nearest neighbor random walk... 

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