Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.
The theory began with the application of BRST gauge fixing procedure to Langevin SDEs, that was later adapted to classical mechanics and its stochastic generalization, higher-order Langevin SDEs, and, more recently, to SDEs of arbitrary form, which allowed to link BRST formalism to the concept of transfer operators and recognize spontaneous breakdown of BRST supersymmetry as a stochastic generalization of dynamical chaos.
The main idea of the theory is to study, instead of trajectories, the SDE-defined temporal evolution of differential forms. This evolution has an intrinsic BRST or topological supersymmetry representing the preservation of topology and/or the concept of proximity in the phase space by continuous time dynamics. The theory identifies a model as chaotic, in the generalized, stochastic sense, if its ground state is not supersymmetric, i.e., if the supersymmetry is broken spontaneously. Accordingly, the emergent long-range behavior that always accompanies dynamical chaos and its derivatives such as turbulence and self-organized criticality can be understood as a consequence of the Goldstone theorem.
History and relation to other theories
The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by Giorgio Parisi and Nicolas Sourlas, who demonstrated that the application of the BRST gauge fixing procedure to Langevin SDEs, i.e., to SDEs with linear phase spaces, gradient flow vector fields, and additive noises, results in N=2 supersymmetric models. The original goal of their work was dimensional reduction, i.e., a specific cancellation of divergences in Feynman diagrams proposed a few years earlier by Amnon Aharony, Yoseph Imry, and Shang-keng Ma. Since then, relation between so-emerged supersymmetry of Langevin SDEs and a few physical concepts have been established including the fluctuation dissipation theorems, Jarzynski equality, Onsager principle of microscopic reversibility, solutions of Fokker–Planck equations, self-organization, etc.
A similar approach was used to establish that classical mechanics, its stochastic generalization, and higher-order Langevin SDEs also have supersymmetric representations. Real dynamical systems, however, are never purely Langevin or classical mechanical. In addition, physically meaningful Langevin SDEs never break supersymmetry spontaneously. Therefore, for the purpose of the identification of the spontaneous supersymmetry breaking as dynamical chaos, the generalization of the Parisi–Sourlas approach to SDEs of general form is needed. This generalization could come only after a rigorous formulation of the theory of pseudo-Hermitian operators because the stochastic evolution operator is pseudo-Hermitian in the general case. Such generalization showed that all SDEs possess N=1 BRST or topological supersymmetry (TS) and this finding completes the story of relation between supersymmetry and SDEs.
In parallel to the BRST procedure approach to SDEs, mathematicians working in the dynamical systems theory introduced and studied the concept of generalized transfer operator defined for random dynamical systems. This concept underlies the most important object of the STS, the stochastic evolution operator, and provides it with a solid mathematical meaning.
STS has a close relation with algebraic topology and its topological sector belongs to the class of models known as Witten-type topological or cohomological field theory. As a supersymmetric theory, BRST procedure approach to SDEs can be viewed as one of the realizations of the concept of Nicolai map.