Chaos, Ito-Stratonovich dilemma, and topological supersymmetry
Abstract
It was recently established that the formalism of the generalized transfer operator (GTO) of dynamical systems (DS) theory, applied to stochastic differential equations (SDEs) of arbitrary form, belongs to the family of cohomological topological field theories (TFT) -- a class of models at the intersection of algebraic topology and high-energy physics. This interdisciplinary approach, which can be called the supersymmetric theory of stochastic dynamics (STS), can be seen as an algebraic dual to the traditional set-theoretic framework of the DS theory, with its algebraic structure enabling the extension of some DS theory concepts to stochastic dynamics. Moreover, it reveals the presence of a topological supersymmetry (TS) in the GTOs of all SDEs. It also shows that among the various definitions of chaos, positive "pressure", defined as the logarithm of the GTO spectral radius, stands out as particularly meaningful from a physical perspective, as it corresponds to the spontaneous breakdown of TS on the TFT side. Via the Goldstone theorem, this definition has a potential to provide the long-sought explanation for the experimental signature of chaotic dynamics known as 1/f noise. Additionally, STS clarifies that among the various existing interpretations of SDEs, only the Stratonovich interpretation yields evolution operators that match the corresponding GTOs and, consequently, have a clear-cut mathematical meaning. Here, we discuss these and other aspects of STS from both the DS theory and TFT perspectives, focusing on links between these two fields and providing mathematical concepts with physical interpretations that may be useful in some contexts.
Summary
This paper explores the intersection of dynamical systems (DS) theory and topological field theories (TFTs) through the lens of stochastic differential equations (SDEs). The main research question revolves around understanding the mathematical structure underlying stochastic dynamics and addressing the Ito-Stratonovich dilemma in interpreting SDEs. The authors introduce the "Supersymmetric Theory of Stochastic Dynamics" (STS), which leverages the generalized transfer operator (GTO) formalism from DS theory and connects it to cohomological TFTs. The methodology involves analyzing the algebraic properties of the GTO, particularly its relationship to topological supersymmetry (TS), and demonstrating how the Stratonovich interpretation of SDEs uniquely aligns with the GTO's mathematical structure. The key findings are that all SDEs possess a topological supersymmetry, and positive "pressure" (related to the GTO's spectral radius) corresponds to the spontaneous breakdown of TS, potentially explaining 1/f noise. The paper also clarifies that only the Stratonovich interpretation of SDEs yields evolution operators consistent with the GTO. This matters to the field because it provides a novel, interdisciplinary framework for studying stochastic dynamics, offering insights into chaos and the interpretation of SDEs. By linking DS theory with TFTs, STS provides a new algebraic perspective that complements traditional set-theoretic approaches, potentially leading to a deeper understanding of complex systems and their stochastic behavior.
Key Insights
- •The formalism of the Generalized Transfer Operator (GTO) applied to stochastic differential equations (SDEs) belongs to the family of cohomological topological field theories (TFTs), establishing a connection between dynamical systems theory and high-energy physics.
- •Positive "pressure", defined as the logarithm of the GTO spectral radius, is identified as a physically meaningful definition of chaos because it corresponds to the spontaneous breakdown of topological supersymmetry (TS).
- •The Stratonovich interpretation of SDEs is uniquely consistent with the GTO, providing a clear-cut mathematical meaning for stochastic evolution operators, resolving the Ito-Stratonovich dilemma.
- •The paper uses the Cartan formula for Lie derivatives to show that the GTO is d-exact (H = [d, d_bar]), a defining characteristic of TFTs, providing a justification for the presence of topological supersymmetry.
- •The paper shows that the GTO has a pseudo-Hermitian structure, leading to a complete bi-orthogonal eigensystem, with complex conjugate pairs of eigenvalues corresponding to Ruelle-Pollicott resonances.
- •The paper proves that zero eigenvalue supersymmetric eigenstate of H^(D), where D is the dimension of the phase space, is always the "ground state" of H^(D), i.e. min Re(spec H^(D)) = 0.
- •The paper provides a STS proof of the stochastic Poincare-Bendixson theorem, stating that spontaneous TS breaking is not possible for models with dimX < 3.
Practical Implications
- •The framework can be used to analyze and model complex systems exhibiting stochastic behavior, such as turbulent flows, financial markets, and biological systems.
- •Researchers in dynamical systems, stochastic modeling, and topological field theory can benefit from the interdisciplinary insights provided by STS.
- •The identification of positive pressure as a signature of chaos and its link to 1/f noise opens up new avenues for understanding and predicting chaotic phenomena.
- •The clarification of the Ito-Stratonovich dilemma offers practical guidance for choosing the appropriate interpretation of SDEs in different applications.
- •Future research directions include exploring the implications of STS for random discrete-time dynamical systems and investigating the role of normalizability in non-compact phase spaces.