Probabilistic potential theory applied to electrical engineering problems

Abstract

The mathematical equivalence between a potential satisfying a deterministic Laplace-type equation within a closed region and a certain probability associated with a particle exercising Brownian motion is described. Two methods are outlined for obtaining a potential by Brownian motion, the usual Monte Carlo method and a "number-diffusion" process. Two probabilistic solutions of Poisson's equation are described. The number-diffusion process with complex diffusion coefficients is applied to a lumped linear electrical network under sinusoidal excitation, for computation of complex voltages. The node equations of a general network are interpreted probabilistically for transient behavior, and computations for a particular case verify the theory. The computation efficiencies (as measured by computing time) of both Monte Carlo and number-diffusion calculations of a potential are compared to the efficiency of matrix manipulation. It is shown how the probabilistic computations at sparse space-time points on a grid of a large number of points in two or three space dimensions may require orders of magnitude less time than for matrix solution, as well as less storage space. Probabilistic solutions of the wave equation for a finite lossy transmission line sinusoidally excited are shown to converge accurately only if |1 + ½(γΔx) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> | > 1 (exp ± γx spacial variation of the natural waves). Then the numerical convergence of the number-diffusion algorithm is examined for various equations and sufficient conditions derived for that convergence.