Supersymmetric Theory of Stochastic ABC Model: A Numerical Study

TL;DR

This paper numerically investigates the stochastic ABC model within the supersymmetric theory of stochastics, revealing spectral properties of the stochastic evolution operator and classifying the system's chaotic behavior.

Contribution

It applies the supersymmetric theory of stochastics to a numerical study of the stochastic ABC model, uncovering spectral characteristics and symmetry properties of the stochastic evolution operator.

Findings

SEO spectra for zeroth and top degree forms do not break supersymmetry

All SDEs possess pseudo-time-reversal symmetry

De Rham cohomology classes correspond to supersymmetric eigenstates

Abstract

In this paper, we investigate numerically the stochastic ABC model, a toy model in the theory of astrophysical kinematic dynamos, within the recently proposed supersymmetric theory of stochastics (STS). STS characterises stochastic differential equations (SDEs) by the spectrum of the stochastic evolution operator (SEO) on elements of the exterior algebra or differentials forms over the system's phase space, X. STS can thereby classify SDEs as chaotic or non-chaotic by identifying the phenomenon of stochastic chaos with the spontaneously broken topological supersymmetry that all SDEs possess. We demonstrate the following three properties of the SEO, deduced previously analytically and from physical arguments: the SEO spectra for zeroth and top degree forms never break topological supersymmetry, all SDEs possesses pseudo-time-reversal symmetry, and each de Rahm cohomology class provides one supersymmetric eigenstate. Our results also suggests that the SEO spectra for forms of complementary degrees, i.e., k and dim X -k, may be isospectral.