Ubiquitous order known as chaos

TL;DR

This work reframes chaos as the spontaneous breakdown of a ubiquitous topological supersymmetry in all stochastic dynamics, introducing the supersymmetric theory of stochastic dynamics ($STS$) as a unifying framework. It develops the mathematical backbone—wavefunctions as differential forms, a stochastic evolution operator ($SEO$), and a cohomological topological field theory ($TFT$) structure—through which the butterfly effect is interpreted as the $TS$-breaking order parameter and long-range dynamical memory. The paper identifies three dynamical regimes ($T$, $N$, $C$), linking 1/f noise to Goldstone-like excitations (goldstinos) and showing how noise-induced instantons generate long-range correlations in the $N$-phase, with numerical demonstrations in a stochastic sine-Gordon model. It also outlines an outlook toward a low-energy effective theory for the $TS$ breaking order parameter and possible holographic duals, suggesting broad implications from neurodynamics to cosmology and offering a rigorous physical foundation for the study of chaos and dynamical information processing. Overall, the work forges a deep connection between dynamical systems and high-energy physics concepts, providing a principled path to a field-theoretic theory of chaos and its practical applications.

Abstract

A close relation has recently emerged between two of the most fundamental concepts in physics and mathematics: chaos and supersymmetry. In striking contrast to the semantics of the word 'chaos,' the true physical essence of this phenomenon now appears to be a spontaneous order associated with the breakdown of the topological supersymmetry (TS) hidden in all stochastic (partial) differential equations, i.e., in all systems from a broad domain ranging from cosmology to nanoscience. Among the low-hanging fruits of this new perspective, which can be called the supersymmetric theory of stochastic dynamics (STS), are theoretical explanations of 1/f noise and self-organized criticality. Central to STS is the physical meaning of TS breaking order parameter (OP). In this paper, we discuss that the OP is a field-theoretic embodiment of the 'butterfly effect' (BE) -- the infinitely long dynamical memory that is definitive of chaos. We stress that the formulation of the corresponding effective theory for the OP would mark the inception of the first consistent physical theory of the BE. Such a theory, potentially a valuable tool in solving chaos-related problems, would parallel the well-established and successful field theoretic descriptions of superconductivity, ferromagentism and other known orders arising from the spontaneous breakdown of various symmetries of nature.

Figures (5)

• Figure 1: (a) The qualitative way to see that chaos is the spontaneous breakdown of topological supersymmetry (TS) is to recall that its definitive property -- the butterfly effect -- is the dynamic separation of initially close points, thereby "breaking" the proximity of points or the topology of the phase space. In the algebraic representation of dynamics, the limit of the long evolution corresponds to the ground state, whereas the "breaking of topology" implies that the ground state is not symmetric with respect to TS, which is the spontaneous symmetry breaking by definition. (b)-(d) Three possible types of spectra of stochastic evolution operator as discussed in text. In (b), the ground state is the zero-eigenvalue supersymmetric eigenstate (black filled circle), indicating unbroken TS. In (c) and (d), the ground states have non-zero eigenvalues, which means that they are non-supersymmetric and TS is spontaneously broken. In (d), the pseudo-time reversal symmetry relating eigenstates with complex conjugate eigenvalues is also broken.
• Figure 2: Numerical examples of the three types of spectra illustrated in Fig.\ref{['figure:1']}b-d in stochastic ABC model on a $D=3$ torus. The spectra are presented separately for each degree, from 3 at the top to 0 at the bottom. The parameters of the model are $A=B=1$, $\Theta=0.83$, and $C=0.8, 1.25, 1$ for a,b, and c, respectively. Each De Rham cohomology class of the phase space provides one supersymmetric eigenstate with zero eigenvalue (black circles at the origin). All other eigenstates are non-supersymmetric pairs that are related by $\hat{d}$ and have degrees differing by unity. They are presented as red circles, green stars, and red squares for degrees 3 and 2, 2 and 1, and 1 and 0, respectively. Red crosses mark the ground states, which, for cases b and c, are non-supersymmetric and consequently doubly degenerate.
• Figure 3: Between the symmetric phase ($T$) with unbroken TS and the conventional chaos ($C$), where TS is broken by the nonintegrability of the constant part of the equations of motion ($f$ in Eq.(\ref{['SDE']})), there exists the phase of noise-induced or intermittent chaos ($N$). In this phase, TS is broken by noise-induced instantons -- the 'quanta' of strongly nonlinear transient dynamics such as earthquakes, neuroavalanches, etc. In the deterministic limit, the $N$-phase collapses into the critical $T$-to-$C$ transition.
• Figure 4: Patterns of dynamics (of $\sin\varphi(r,t)$) in the overdamped sine-Gordon equation with non-potential driving on a spatial circle of length 300, which can be viewed as a coarse-grained chain of type-I neurons. In the deterministic limit, the T-C transition (see Fig.\ref{['figure:3']}) occurs at $\alpha=1$, when vacuum at $\varphi = \sin^{-1}\alpha$ loses stability. Accordingly, Figs. (a)-(c), with parameters $\alpha= 0.92, 0.97$, and $1.1$ (and $\Theta=0.03$), correspond respectively to the $T-$, $N-$, and $C-$ phases as evident from Fig.\ref{['figure:5']}a. The dynamics in the N-phase (b) is dominated by noise-induced instantonic processes of creation and annihilation of pairs of solitons which are the left/right moving kinks/antikinks. In the context of neurodynamics, the kink-antikink pairs can be seen as one-dimensional predecessors of neuroavalanches.
• Figure 5: (a) The part of the phase diagram of the overdamped sine-Gordon equation with non-potential driving Eq.(\ref{['SGE']}), constructed by maximal stochastic Lyapunov exponents. Grey and colored dots represent negative and positive exponents, respectively. (b) A qualitative illustration of the essence of the butterfly effect in the $N$-phase of the model. Near and slightly below $\alpha=1$, even a weak perturbation can introduce a kink-antikink pair, which, due to interaction with other kink and antikinks, forms two "quasi-particles" (dashed curves) of the difference between the perturbed (red) and unperturbed (green) dynamical patterns. These quasi-particles can live for an infinitely long time, thereby manifesting the presence of the butterfly effect.