Topological supersymmetry breaking: The definition and stochastic generalization of chaos and the limit of applicability of statistics

Abstract

The concept of deterministic dynamical chaos has a long history and is well established by now.Nevertheless, its field theoretic essence and its stochastic generalization have been revealed only very recently. Within the newly found supersymmetric theory of stochastics (STS), all stochastic differential equations (SDEs) possess topological or de Rahm supersymmetry and stochastic chaos is the phenomenon of its spontaneous breakdown. Even though the STS is free of approximations and thus is technically solid, it is still missing a firm interpretational basis in order to be physically sound. Here, we make a few important steps toward the construction of the interpretational foundation for the STS. In particular, we discuss that one way to understand why the ground states of chaotic SDEs are conditional (not total) probability distributions, is that some of the variables have infinite memory of initial conditions and thus are not "thermalized", i.e., can not be described by the initial-conditions-independent probability distributions. As a result, the definitive assumption of physical statistics that the ground state is a steady-state total probability distribution is not valid for chaotic SDEs. * iovchinnikov@ucla.edu

Criticality or Supersymmetry Breaking?

“…Therefore, when TS is broken spontaneously, Γ g together with S are positive, and it immediately follows that the phenomenon of the spontaneous breakdown of TS must be associated with the stochastic generalization of dynamical chaos [42].…”

“…In other words, TS is the preservation of the proximity of points in phase space during evolution. This interpretation makes it particularly clear that the spontaneous breakdown of TS must be viewed as the stochastic generalization of dynamical chaos [42,43]. Indeed, the spontaneous breakdown of TS must imply that initially close points may not be close anymore after an infinitely long evolution, when the system is described by a non-supersymmetric ground state.…”

Stochastic dynamics and combinatorial optimization

“…It can be shown that such situation can be looked upon as the stochastic generalization of the concept of deterministic chaos. [9] …”

“…[8] This theory is an approximation-free and coordinate-free theory of SDEs that can be dubbed the supersymmetric theory of stochastics (STS) and that has provided a rigorous stochastic generalization of the concept of deterministic dynamical chaos. [9] Another important result from the STS, [10] which is directly relevant to this paper, is revealing the theoretical essence of the stochastic dynamical behavior known as noise-induced chaos, [11] self-organized criticality, [7] intermittency [12,13] etc. and that we call for brevity the N-phase.…”

Dynamical Field Inference and Supersymmetry

“…The approach of supersymmetric theory of stochastics (STS) focuses on the theoretical analysis of the DS as a supersymmetric system [ 18 , 19 , 20 , 21 ]. One of the central messages of stochastics (STS) is the correspondence between the spontaneous breakdown of this supersymmetry (SUSY) and the emergence of chaotic dynamics.…”