Field-theoretic methods

TL;DR

This paper discusses how field-theoretic methods, using functional integrals, can analyze complex spatio-temporal systems by mapping stochastic processes onto continuum field theories, enabling systematic computation of correlation functions.

Contribution

It provides a unified framework for applying field-theoretic techniques to nonlinear stochastic processes, including particle systems and Langevin equations, with explicit mappings and illustrative examples.

Findings

Functional integral techniques facilitate correlation function computation.

Mappings connect stochastic processes to continuum field theories.

Methodology is illustrated with well-studied examples.

Abstract

Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that facilitates the computation of such correlation functions and moments, and furthermore allows a systematic development of perturbation expansions and other useful approximative schemes. It is explained how nonlinear stochastic processes may be mapped onto exponential probability distributions, whose weights are determined by continuum field theory actions. Such mappings are madeexplicit for (1) stochastic interacting particle systems whose kinetics is defined through a microscopic master equation; and (2) nonlinear Langevin stochastic differential equations which provide a mesoscopic description wherein a separation of time scales between the relevant degrees of freedom and background statistical noise is assumed. Several well-studied examples are introduced to illustrate the general methodology.