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

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.