Chaos, Ito-Stratonovich dilemma, and topological supersymmetry

TL;DR

This work develops and consolidates the supersymmetric theory of stochastic dynamics (STS), unifying dynamical systems theory with cohomological topological field theories via the generalized transfer operator (GTO) and its topological supersymmetry (TS). It argues that chaos corresponds to spontaneous TS breaking and that the Stratonovich interpretation is essential for a mathematically consistent stochastic evolution operator (SEO) that matches the GTO, with implications for explaining 1/f noise through Goldstone-type modes. The formalism employs path-integral methods, BRST gauge fixing, and Morse–Smale–Witten constructs to relate instantaneous transitions (instantons) to topological invariants, yielding a phase diagram that separates integrable, non-integrable, and noise-dominated regimes. Overall, STS provides a rigorous, topological perspective on stochastic dynamics, offering theoretical explanations for chaotic signatures and framing a bridge between DS theory and TFT that may inspire cross-disciplinary insights.

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.

Figures (5)

• Figure 1: (left) A continuous-time stochastic DS is defined by a flow vector field, $\mathscr F$, from the tangent space, $TX$, of the phase space, $X$. ${\mathscr F}(\xi(t))$ is time-dependent due to the presence of the time-dependent noise, $\xi(t)$. The DS is equivalent to two-parameter family of noise-configuration-dependent diffeomorphisms, $M(\xi)_{tt'}$, such that the trajectories are given by $x(t)=M(\xi)_{tt'}(x(t'))$. (right) In the spirit of the pathintegral representation of temporal evolution, there is a copy of $X$ at every time moment and evolution is defined by pullbacks, $\hat{M}(\xi)_{t't}^*$, induced by inverse maps, $M(\xi)_{t't}$. The pullbacks act on a time-dependent differential form, $|\psi(t)\rangle$, understood as a "wavefunction" - a time-dependent object encoding information of the system's past. When averaged over noise configurations, a pullback yields the generalized transfer operator, which is unique and corresponds to the Stratonovich interpretation of stochastic dynamics.
• Figure 2: The three possible types of GTO spectra (a,b,c) for a stochastic DS with $X=\mathbb{S}^3$. Each row ($k = 0 , . . .3$) contains three graphs representing $\text{spec}H^{(k)}$ for the three different types of spectra. Dots at the origin for $k=0$ and $k=3$ indicate the supersymmetric eigenstates from the zeroth and the third de Rham cohomologies of $X$. In cases b and c, the ground states (crosses) are non-supersymmetric doublets, as they possess non-zero eigenvalues, signifying spontaneous breakdown of TS. Additionally, in case c, the pseudo-time reversal symmetry is also broken. The vertical arrowed curves illustrate the action of the TS operator.
• Figure 3: Pathintegral is a continuous-time limit, $N\to\infty, \Delta \tau = (t-t')/N\to0$, of the discrete time evolution picture: the domain of temporal evolution, $(t,t')$, is split into $N$ time steps and the time takes on discrete values $\tau_N, \tau_{N-1},...\tau_1,\tau_0$, $t=\tau_N, t'=\tau_0$. Each time slice hosts a supersymmetric pair of fields $x_k, \chi_k$, and each dual slice hosts the corresponding supersymmetric pair of momenta fields, $B_k, \bar{\chi}_k$, along with the noise variable, $\xi_k$. The finite-time stochastic evolution operator is derived by integrating out all the fields except those at the first and last slices. Its exact expression depends on parameter $\alpha\in(0,1)$, which dictates how $x$ and $\chi$ are interpreted at the dual slice, $\tau_k$: $\alpha x_{k} + (1-\alpha)x_{k-1}$. Only for $\alpha=1/2$, corresponding to the Stratonovich interpretation of SDEs, does the stochastic evolution operator matches the generalized transfer operator of the DS theory, thereby having a clear-cut mathematical meaning of the pullback averaged over noise.
• Figure 4: An example of a Morse-Smale flow (thin green arrowed curves). The filled green circles (b, e) represent minima (index 0), hollow circles (a,f,g,h) correspond to saddles (index-1), and filled-black circles (c,b) denote index-2 critical points. The bras/kets of the local supersymmetric states of the Morse-Smale-Bott-Witten complex are Poincare duals of the local stable/unstable manifolds. For example, $\langle a| = p(S_a)$ and $|a\rangle=p(U_a)$ are narrow distributions on $S_a=(cad)$ and $U_a=(eab)$, respectively, with fermions in transverse directions, whereas $\langle b|$ and $|d\rangle$ are constant functions over the green and gray regions, respectively. The dashed curves represent the one-parameter families of 1-dimensional manifolds, $\gamma_1(t), \gamma_2(t)$, obtained by the flow-defined diffeomorphisms, $\gamma_1(t) = M_{t0}(\gamma_1)$. Their Poincare duals can be used to construct, e.g, the matrix element, $\langle b| \hat{p}(\gamma_1(t))\hat{p}(\gamma_2(0))|d\rangle =1$, which represents the intersection number of $\gamma$'s on the instanton manifold, $I_{bd} = S_b \cap U_d =(bhda)$. The matrix element is independent of $t$'s because the intersection points (dis)appear in pairs with opposite orientations (white and black filled circles).
• Figure 5: Stochastic DSs can be classified based on two key factors: (i) whether the topological supersymmetry (TS) is spontaneously broken (ordered phase) or unbroken (symmetric phase) and (ii) whether the flow vector field is integrable or non-integrable and/or chaotic. The symmetric phase with unbroken TS is labeled as T. The ordered phase with non-integrable flow (C-phase) is a stochastic generalization of the deterministic chaos, where the TS breaking is caused by the nonintegrability of the flow. The ordered phase with integrable flow (N-phase) can be identified as the noise-induced chaos, where the dynamics is dominated by noise-induced instantons. The instantons vanish in the deterministic limit, causing the N-phase to collapse onto the boundary of the C-phase. As noise intensity increases, the TS must eventually be restored disregard of the properties of the flow, as the GTO/SEO becomes dominated by the Laplacian, which alone does not break TS.