Random Walk and Brownian Motion

Abstract

Random walk processes are an important class of stochastic processes. They have many applications in physics, computer science, ecology, economics, and other fields. A random walk is a sequence of successive random steps. In this chapter we study Markovian discrete time models. The time evolution of a system is described in terms of a N-dimensional vector, which can be, for instance, the position of a molecule in a liquid or the price of a fluctuating stock. In a first experiment we compare one-dimensional walks with constant or varying step sizes. The mean square distance is compared to the central limit theorem. A three-dimensional walk is used to simulate the freely jointed chain model in polymer physics. In a computer experiment we calculate the gyration tensor which is relevant to scattering experiments. We discuss the simplified Hookean spring model to simulate the force–extension relation. In a further experiment we study Brownian motion in a harmonic potential.