Supersymmetric theory of stochastic ABC model

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 characterizes stochastic differential equations (SDEs) by the spectrum of the stochastic evolution operator (SEO) on elements of the exterior algebra or differential 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 SEOs of 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 possess pseudo-time-reversal symmetry, and each de Rham cohomology class provides one supersymmetric eigenstate. Our results also suggest that the SEO spectra for forms of complementary degrees, i.e., k and dim X - k, may be isospectral.