Estudo da difusão de partículas sob influência de um campo laplaciano

Abstract

In this work, we study the traditional Random Walk model and some of its variations, performing several computer simulations in two and three dimensions. We calculate the mean squared displacement in order to analyze the dispersion of the particles, namely, how much they move away from the origin, and observe their linear or non-linear characteristic with time, and how such behavior depends on the restrictions adopted in the movement of the particles. We apply a variation on the traditional Random Walk model introducing the condition that the particle cannot occupy a region in the space already occupied by it, thus forming a Self-AvoidingWalk. And, in order to obtain a Self-Avoiding Walk without trapping in its structural configurations, we use the so-called model Laplacian Random Walk. Where the random movement of the particles is coupled to a Laplacian field. The probability of direction of motion depend on the solution of the Laplace Equation, causing the particle to recognize possible trappings in advance, thus being able to avoid them. We observe that particles performing a Self-Avoiding Walk or a Laplacian Random Walk behave in a superlinear way and we analyze how this affects the mean squared displacement, the diffusion coefficient and the fractal dimension of the system.