Accommodation and Void Distribution in the Random Walk.

The accommodation of particles in the random walk and the void distribution in the random walk are studied. In the unrestricted random walk particle accommodation is studied using the diffusion equation. Calculations in the two, three and arbitrary dimensions shows that the accommodation as a function of the particle size behave as A(r) ~ r^{d - 3}, where d is the embedding dimension of the walk. The numerical evaluation of the accommodation is in agreement with this relation. This result is suggestive of an accommodation that depends on fractal dimension D of the host (e.g. random walk) in the following fashion A(r) ~ r^{d - D - 1}. A computer simulation of accommodation of particles in the self-avoiding random walk in the two dimensions supports this conjecture. A calculation of the perimeter-area relation using a computer, is found to follow a scaling law which is characteristic of a self-avoiding polygon. The distribution of the closed voids in the two dimensional RW is calculated by comparing the accommodation function for the RW and the accommodation in the closed voids of a fractal. This distribution is found to be n(v ) ~ v^ {- 2.2} when the fractal nature of the perimeter of the voids is taken onto account. Calculation using a computer gives an exponent which is in close agreement with this distribution.