Stochastic Processes, Slaves and Supersymmetry

TL;DR

This paper explores the supersymmetry structure of diffusion processes by analyzing how external disturbances affect stochastic systems, introducing slave variables, and examining their statistical properties, especially at low temperatures.

Contribution

It extends previous work by incorporating slave variables to understand the response of stochastic systems and analyzes their supersymmetry properties in a three-dimensional electromagnetic analogy.

Findings

Slave variable distributions lose variance below a critical temperature.

Supersymmetry properties help in calculating asymptotic low-temperature results.

Numerical simulations reveal temperature-dependent behavior of slave variables.

Abstract

We extend the work of Tanase-Nicola and Kurchan on the structure of diffusion processes and the associated supersymmetry algebra by examining the responses of a simple statistical system to external disturbances of various kinds. We consider both the stochastic differential equations (SDEs) for the process and the associated diffusion equation. The influence of the disturbances can be understood by augmenting the original SDE with an equation for {\it slave variables}. The evolution of the slave variables describes the behaviour of line elements carried along in the stochastic flow. These line elements together with the associated surface and volume elements constructed from them provide the basis of the supersymmetry properties of the theory. For ease of visualisation, and in order to emphasise a helpful electromagnetic analogy, we work in three dimensions. The results are all generalisable to higher dimensions and can be specialised to one and two dimensions. The electromagnetic analogy is a useful starting point for calculating asymptotic results at low temperature that can be compared with direct numerical evaluations. We also examine the problems that arise in a direct numerical simulation of the stochastic equation together with the slave equations. We pay special attention to the dependence of the slave variable statistics on temperature. We identify in specific models the critical temperature below which the slave variable distribution ceases to have a variance and consider the effect on estimates of susceptibilities.