Mean 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... 

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... 

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... 

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

Figure 4.46. The mean walklength n)(C/P) versus the lattice edge length i for a target molecule anchored at the center of a face (solid line), the midpoint of an edge (dashed line), or a vertex position (hyphenated line). The governing potential is an attractive ion-ion potential with W = -1.

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