Walklengths

In Section III the temporal behavior of diffusion-reaction processes occurring in or on compartmentalized systems of various geometries, as determined via solution of the stochastic master equation (4.3), is studied. Also, in Sections III-V, results are presented for the mean walklength (n). From the relation (4.7), and the structure of the solutions (4.6) to Eq. (4.3), the reciprocal of (n) may be understood as an effective first-order rate constant k for the process (4.2) or (n) itself as a measure of the characteristic relaxation time of the system it is, in effect, a signature of the long-time behavior of the system. 

Two strategies that can be used to simplify the calculation of the mean walklength n) are now reviewed. In the first of these, it is shown that if the random walk is modeled by a stationary Markov process on a finite state... 

The answers to these questions are furnished by the matrix [I — P] , which is denoted here as II. II exists if and only if the expected walklength from each possible starting point is finite. The k, j)th element of this matrix is just the quantity n) j referred to in the previous paragraph. Although there are some interesting questions not answered by knowledge of the matrices P and II alone, such as the distributions of the numbers of visits to various sites, rather than just the expected values of these numbers, attention in this review is limited to the (numerous) properties of random walks that can be derived from knowledge of these two matrices. 

To illustrate the advantages gained in considering lattice symmetries, consider a target molecule B (or trap) positioned at an arbitrary site on a finite, 5x5 square-planar lattice. Calculation of the mean walklength ( ) before reaction (trapping) of a coreactant A diffusing on this lattice, and subject to specific boundary conditions, requires the specification of the matrix P and subsequent inversion of the matrix [I — P], If the trap is anchored at the centrosymmetric site on the lattice and periodic boundary... 

In terms of the notation introduced above, (11)33 = 6.4, n) =32, and the overall mean walklength of the random walker before being trapped is... 

Figure 4.2. Average walklength (n) versus system size iV for d = 2 and v = 3,4,6.

To get a preliminary handle on the kind of differences that arise when changes in dimensionality and/or valency are made, consider a fivefold expansion in the total number N of sites for a lattice of given d,v), that is, consider values of the average walklength [n) at N = 50 and N — 250. Suppose one fixes the dimensionality of the lattice and considers the change... 

Figure 4.5. Average walklength n) versus system size N for v = 6 and d = 2.3.

It is of interest to compare the numerically exact results presented in refs. 9,10, and 13 for dimension d = 2 with the asymptotic analytic results obtained by Montroll and Weiss [17-19]. In this classic study, these authors showed that the overall average walklength (n) can be expressed explicitly... 

Site specific walklengths ( >, for lattices of a common valency v = 6) but different dimensionalities (see text)... 

Comparison of two analytic representations of the data for the mean walklength n) on a finite, cubic lattice with a centrosymmetric trap and subject to periodic boundary conditions... 

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... 

Consider now the lattices diagrammed in Figures 4.6 and 4.7. Whereas the N dependence of (n) will be found to be qualitatively correct, the fact that these finite lattices are not characterized by a uniform valency v limits the usefulness of Eq. (4.21). In fact, it is necessary to introduce composite (average) lattice parameters in order to be able to compare trends in the data. Specifically, one parameter is needed to account for the nonuniform valency and one parameter to reflect and/or characterize the variety of geometrical shapes. Consistent with the identification of an overall (average) walklength... 

Values of the mean walklength n) for lattices of integer dimension... 

Figure 4.19. A plot of the average walklength ( ) versus system size (normalized edge length ) for N X N X N simple cubic lattices. The curve through the filled circles gives the results for a d = 3 walk to a central trap. The curve through the open squares displays the results obtained using the tracking boundary condition (see text), with the trap anchored at a centrosymmetric site on the boundary of the compartmentalized (lattice) system.

Figure 4.22. A plot of the average walklength (n) versus system size (normalized edge length i) for a series of 10 x 10 x A lattices. The conventions here are the same as in Figure 4.21.

This definition of the spectral dimension agrees with that of Eq. (4.41) in the limit as n goes to infinity. The numerical simulation of the spectral dimension based on random-walk models requires enormously large lattices and long walklengths to obtain reliable results [59]. 

Instead of mobilizing large scale Monte Carlo simulations of the visitation probability P as a function of walklength n, Pk can be evaluated (as a function of time) using the stochastic master equation [60]. Suppose at time f = 0 a random walker is positioned with unit probability at a site m in the interior of the lattice (away from the boundary of the system). For t > 0 this probability evolves among the lattice sites as determined by Eq. (4.3). An entropy-like quantity... 

The second trend is that, for a fixed setting of v, both the average walklength n) and the relaxation time increase with increase in N. See the data on the three figures for which u = 5, the four figures for which v = 4, and the seven structures displayed in Figures 4.28 and 4.29 in the case v — 3. 

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