Formalismo da integral funcional para dinâmica estocástica multidimensional

Abstract

We present a study of stochastic processes that describe a variety of physical phenomena such as Brownian motion, described by a stochastic phenomenological equation, called the Langevin equation. This equation contains two characteristic forces: a frictional and a random force called noise. With this type of stochastic equations, with multiplicative noise, arise di erent prescriptions to integrate the noise (or to discretize of the di erential equation). Each of them produces a di erent stochastic evolution associated to particular calculation rules. These prescriptions are related to the so called Wiener process. The multiplicity of possible prescriptions to de ne mathematically these processes is an obstacle to the development of general tools for their treatment. For this reason we used a tool called formulation of functional integral. This formulation was developed in detail for a random variable in Ref.(13). The main purpose of this dissertation is to extend this formalism to the case of a multidimensional system of Langevin equations with multiplicative noise. We deduced an path integral on a set of fermionic and bosonic variables without performing any discretization. The usual prescriptions to de ne the Wiener integral arise in the formalism by the de nitions of Green functions in the sector of Grassmann variables in the eld theory. Finally we obtain a dynamic generator functional which is an important tool to describe physical phenomena such as phase transitions induced by noise.