A Closed General Solution of the Probability Distribution Function for Three-Dimensional Random Walk Processes

Abstract

A closed general solution of the probability distribution function for three‐dimensional random walk processes is derived. In addition: (1) For the particular case of equal‐length displacements, the exact solution is compared with the Gaussian approximation for n=3, 5, and 10 steps. (2) The general solution is utilized in calculating the probability distribution of gamma‐ray energies resulting in the Cl35 (n, γ) Cl36 process. (3) For five unequal steps of fractional length: 0.582, 0.135, 0.131, 0.092, and 0.060 (which is somewhat characteristic of the fractional energies of gamma rays resulting from neutron capture), the exact solution is compared with a Gaussian, a modified Gaussian, and a five equal‐step approximation. (4) There are presented the specific solutions for all possible unequal‐length random displacements involving n=2, 3, and 4 steps