Langevin Equations with Multiplicative Noise: Theory and Applications to Physical Processes

Abstract

This chapter discusses a certain type of stochastic differential equation (SDE) that is quite useful for heuristic modeling of physical processes. In much of the literature, particularly in the physical sciences, such equations are usually referred to as Langevin equations, after P. Langevin, who was the first to use such equations to model the dynamics of physical systems. To model the dynamics of a physical process by a SDE, the origin and properties of the fluctuations must be identified. To discuss the origin of the fluctuations, it is often convenient to introduce the idealization of a closed physical system to distinguish between internal and external fluctuations. The microscopic dynamics of the closed system are determined by a Hamiltonian so that the knowledge of all the degrees of freedom at one instant of time is sufficient to determine the evolution of the system forward in time.