Working field theory problems with random walks

Abstract Purpose – The purpose of this paper is to demonstrate how Monte Carlo methods can be applied to the solution of field theory problems. Design – This objective is achieved by building insight from Laplacian field problems. The point solution of a Laplacian field problem can be viewed as the solid angle average of the Dirichlet potentials from that point. Alternatively it can be viewed as the average of the termination potential of a number of random walks. Poisson and Helmholtz equations add the complexity of collecting a number of packets along this walk, and noting the termination of a random walk at a Dirichlet boundary. Findings – When approached as a Monte Carlo problem, Poisson type problems can be interpreted as collecting and summing source packets representative of current or charge. Helmholtz problems involve the multiplication of packets of information modified by a multiplier reflecting the conductivity of the medium. Practical implications – This method naturally lends itself to parallel processing computers. Originality/value – This is the first paper to explore random walk solutions for all classes of eddy current problems, including those involving velocity. In problems involving velocity, the random walk direction enters depending on the walk direction with respect to the local velocity. Keywords Finite difference methods, Monte Carlo methods, Random functions Paper type Technical paper