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Supersymmetric theory of stochastic dynamics
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Supersymmetric theory of stochastic dynamics
Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic dynamics on the intersection of dynamical systems theory, topological field theories, stochastic differential equations (SDE), and the theory of pseudo-Hermitian operators. It can be seen as an algebraic dual to the traditional set-theoretic framework of the dynamical systems theory, with its added algebraic structure and an inherent topological supersymmetry (TS) enabling the generalization of certain concepts from deterministic to stochastic models.
Using tools of topological field theory originally developed in high-energy physics, STS seeks to give a rigorous mathematical derivation to several universal phenomena of stochastic dynamical systems. Particularly, the theory identifies dynamical chaos as a spontaneous order originating from the TS hidden in all stochastic models. STS also provides the lowest level classification of stochastic chaos which has a potential to explain self-organized criticality.
The traditional approach to stochastic dynamics focuses on the temporal evolution of probability distributions. At any moment, the distribution encodes the information or the memory of the system's past, much like wavefunctions in quantum theory. STS uses generalized probability distributions, or "wavefunctions", that depend not only on the original variables of the model but also on their "superpartners", whose evolution determines Lyapunov exponents. This structure enables an extended form of memory that includes also the memory of initial conditions/perturbations known in the context of dynamical chaos as the butterfly effect.
From an algebraic topology perspective, the wavefunctions are differential forms and dynamical systems theory defines their dynamics by the generalized transfer operator (GTO)—the pullback averaged over noise. GTO commutes with the exterior derivative, which is the topological supersymmetry (TS) of STS.
The presence of TS arises from the fact that continuous-time dynamics preserves the topology of the phase/state space: trajectories originating from close initial conditions remain close over time for any noise configuration. If TS is spontaneously broken, this property no longer holds on average in the limit of infinitely long evolution, meaning the system is chaotic because it exhibits a stochastic variant of the butterfly effect. In modern theoretical nomenclature, chaos, along with other realizations of spontaneous symmetry breaking, is an ordered phase—a perspective anticipated in early discussions of complexity: as pointed out in the context of STS:
The Goldstone theorem necessitates the long-range response, which may account for 1/f noise. The Edge of Chaos is interpreted as noise-induced chaos—a distinct phase where TS is broken in a specific manner and dynamics is dominated by noise-induced instantons. In the deterministic limit, this phase collapses onto the critical boundary of conventional chaos.
The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by Giorgio Parisi and Nicolas Sourlas, where Langevin SDEs—SDEs with linear phase spaces, gradient flow vector fields, and additive noises—were given supersymmetric representation with the help of the BRST gauge fixing procedure. While the original goal of their work was dimensional reduction, the so-emerged supersymmetry of Langevin SDEs has since been addressed from a few different angles including the fluctuation-dissipation theorems, Jarzynski equality, Onsager principle of microscopic reversibility, solutions of Fokker–Planck equations, self-organization, etc.
The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including classical mechanics, its stochastic generalization, and higher-order Langevin SDEs. The theory of pseudo-Hermitian supersymmetric operators and the relation between the Parisi-Sourlas method and Lyapunov exponents further enabled the extension of the theory to SDEs of arbitrary form and the identification of the spontaneous BRST supersymmetry breaking as a stochastic generalization of chaos.
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Supersymmetric theory of stochastic dynamics
Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic dynamics on the intersection of dynamical systems theory, topological field theories, stochastic differential equations (SDE), and the theory of pseudo-Hermitian operators. It can be seen as an algebraic dual to the traditional set-theoretic framework of the dynamical systems theory, with its added algebraic structure and an inherent topological supersymmetry (TS) enabling the generalization of certain concepts from deterministic to stochastic models.
Using tools of topological field theory originally developed in high-energy physics, STS seeks to give a rigorous mathematical derivation to several universal phenomena of stochastic dynamical systems. Particularly, the theory identifies dynamical chaos as a spontaneous order originating from the TS hidden in all stochastic models. STS also provides the lowest level classification of stochastic chaos which has a potential to explain self-organized criticality.
The traditional approach to stochastic dynamics focuses on the temporal evolution of probability distributions. At any moment, the distribution encodes the information or the memory of the system's past, much like wavefunctions in quantum theory. STS uses generalized probability distributions, or "wavefunctions", that depend not only on the original variables of the model but also on their "superpartners", whose evolution determines Lyapunov exponents. This structure enables an extended form of memory that includes also the memory of initial conditions/perturbations known in the context of dynamical chaos as the butterfly effect.
From an algebraic topology perspective, the wavefunctions are differential forms and dynamical systems theory defines their dynamics by the generalized transfer operator (GTO)—the pullback averaged over noise. GTO commutes with the exterior derivative, which is the topological supersymmetry (TS) of STS.
The presence of TS arises from the fact that continuous-time dynamics preserves the topology of the phase/state space: trajectories originating from close initial conditions remain close over time for any noise configuration. If TS is spontaneously broken, this property no longer holds on average in the limit of infinitely long evolution, meaning the system is chaotic because it exhibits a stochastic variant of the butterfly effect. In modern theoretical nomenclature, chaos, along with other realizations of spontaneous symmetry breaking, is an ordered phase—a perspective anticipated in early discussions of complexity: as pointed out in the context of STS:
The Goldstone theorem necessitates the long-range response, which may account for 1/f noise. The Edge of Chaos is interpreted as noise-induced chaos—a distinct phase where TS is broken in a specific manner and dynamics is dominated by noise-induced instantons. In the deterministic limit, this phase collapses onto the critical boundary of conventional chaos.
The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by Giorgio Parisi and Nicolas Sourlas, where Langevin SDEs—SDEs with linear phase spaces, gradient flow vector fields, and additive noises—were given supersymmetric representation with the help of the BRST gauge fixing procedure. While the original goal of their work was dimensional reduction, the so-emerged supersymmetry of Langevin SDEs has since been addressed from a few different angles including the fluctuation-dissipation theorems, Jarzynski equality, Onsager principle of microscopic reversibility, solutions of Fokker–Planck equations, self-organization, etc.
The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including classical mechanics, its stochastic generalization, and higher-order Langevin SDEs. The theory of pseudo-Hermitian supersymmetric operators and the relation between the Parisi-Sourlas method and Lyapunov exponents further enabled the extension of the theory to SDEs of arbitrary form and the identification of the spontaneous BRST supersymmetry breaking as a stochastic generalization of chaos.