Path integral methods for the dynamics of stochastic and disordered systems

TL;DR

This paper reviews path integral techniques for analyzing the dynamics of disordered systems under thermal and quenched noise, covering formalism, supersymmetry, and applications to soft and hard spin models.

Contribution

It provides a comprehensive pedagogical overview of path integral methods, including supersymmetric formulations and recent advances in kinetic Ising model dynamics.

Findings

Diagrammatic approach to stochastic dynamics

Supersymmetry implications for Langevin dynamics

Recent developments in kinetic Ising models

Abstract

We review some of the techniques used to study the dynamics of disordered systems subject to both quenched and fast (thermal) noise. Starting from the Martin-Siggia-Rose path integral formalism for a single variable stochastic dynamics, we provide a pedagogical survey of the perturbative, i.e. diagrammatic, approach to dynamics and how this formalism can be used for studying soft spin models. We review the supersymmetric formulation of the Langevin dynamics of these models and discuss the physical implications of the supersymmetry. We also describe the key steps involved in studying the disorder-averaged dynamics. Finally, we discuss the path integral approach for the case of hard Ising spins and review some recent developments in the dynamics of such kinetic Ising models.