Supersymmetric theory of stochastic dynamics

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Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic dynamics on the intersection of dynamical systems theory, statistical physics, stochastic differential equations (SDE), topological field theories, and the theory of pseudo-Hermitian operators. The theory can be viewed as a generalization of the Parisi-Sourlas method to SDEs of arbitrary form or as an adaptation of the concept of the generalized transfer operator of dynamical systems theory to stochastic dynamics.

The main idea of the theory is to examine the temporal evolution of differential forms, with their differentials encoding the dynamical memory associated with Lyapunov exponents. This evolution has an intrinsic topological supersymmetry, whose spontaneous breakdown is the stochastic generalization of chaos. Through the Goldstone theorem, STS offers a potential explanation for universal long-range phenomena such as 1/f, flicker, and crackling noises and the power-law statistics of instantonic processes like earthquakes and neuroavalanches.

The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by Giorgio Parisi and Nicolas Sourlas,^[1][2] 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. The original goal of their work was dimensional reduction, i.e., a specific cancellation of divergences in Feynman diagrams proposed a few years earlier by Amnon Aharony, Yoseph Imry, and Shang-keng Ma.^[3] Since then, the so-emerged supersymmetry of Langevin SDEs has been addressed from a few different angles ^{[4][5][6][7][8]} including the fluctuation dissipation theorems,^[7] Jarzynski equality,^[9] Onsager principle of microscopic reversibility,^[10] solutions of Fokker–Planck equations,^[11] self-organization,^[12] etc.

The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including classical mechanics,^[13][14] its stochastic generalization,^[15] and higher-order Langevin SDEs^[8]. The development of the theory of pseudo-Hermitian supersymmetric operators ^[16] further enabled its generalization to SDEs of arbitrary form ^[17]. The universal nature of this supersymmetry, along with its connection to Lyapunov exponents ^[18], necessitates identifying its spontaneous breakdown as a stochastic generalization of chaos.

In parallel, mathematicians in the dynamical systems theory introduced the concept of the generalized transfer operator defined for random dynamical systems.^[19][20] This concept underlies the stochastic evolution operator of STS and provides it with a solid mathematical meaning. Similar constructions were studied in the theory of SDEs^{[21] [22]}.

The Parisi-Sourlas method has been recognized ^[23][13] as a member of Witten-type or cohomological topological field theory^{[24][25] [26][27][28][29][30][31]}, a class of models to which STS also belongs.

A continuous-time non-autonomous dynamical system can be defined as, $${\displaystyle {\dot {x}}(t)=F(x(t))+(2\Theta )^{1/2}G_{a}(x(t))\xi ^{a}(t)\equiv {\mathcal {F}}(\xi (t)),}$$ where ${\textstyle x\in X}$ is a point in the phase space which can be assumed to be a closed smooth manifold, ${\displaystyle F(x)\in TX_{x}}$ is a sufficiently smooth flow vector field from the tangent space of ${\displaystyle X}$, and ${\displaystyle G_{a}\in TX,a=1,\ldots ,D,D=\dim X}$ is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, ${\displaystyle \xi (t)\in \mathbb {R} ^{D}}$, which is called additive/multiplicative depending on whether ${\displaystyle G_{a}}$'s are independent/dependent on the position on ${\displaystyle X}$. The randomness of the noise will be introduced later. For now, the noise is a deterministic function of time and the equation above is an ordinary differential equation (ODE) with a time-dependent flow vector field, ${\displaystyle {\mathcal {F}}}$.

This ODE defines a unique solution/trajectory, ${\displaystyle x(t)}$, for any initial condition, ${\displaystyle x(t')=x'}$. Even for noise configurations that are non-differentiable with respect to time, ${\displaystyle t}$, the solution is differentiable with respect to the initial condition, ${\displaystyle x'}$.^[32] In other words, there exists a two-parameter family of noise-configuration-dependent diffeomorphisms: $${\displaystyle M(\xi )_{tt'}:X\to X,M(\xi )_{tt'}\circ M(\xi )_{t't''}=M(\xi )_{tt''},\left.M(\xi )_{tt'}\right|_{t=t'}={\text{Id}}_{X},}$$ such that the above solution of the ODE can be expressed as ${\displaystyle x(t)=M(\xi )_{tt'}(x')}$.

The dynamics can now be defined as follows: if at time ${\displaystyle t'}$, the system is described by the probability distribution ${\displaystyle P(x)}$, then the average value of some function ${\displaystyle f:X\to \mathbb {R} }$ at a later time ${\displaystyle t}$ is given by: $${\displaystyle {\bar {f}}(t)=\int _{X}f\left(M(\xi )_{tt'}(x)\right)P(x)dx^{1}\wedge ...\wedge dx^{D}=\int _{X}f(x){\hat {M}}(\xi )_{t't}^{*}\left(P(x)dx^{1}\wedge ...\wedge dx^{D}\right).}$$ Here ${\displaystyle {\hat {M}}(\xi )_{t't}^{*}}$ is action or pullback induced by the inverse map, ${\displaystyle M(\xi )_{tt'}^{-1}=M(\xi )_{t't}}$, on the probability distribution understood in a coordinate-free setting as a top-degree differential form.

Pullbacks are a wider concept, defined also for k-forms, i.e., differential forms of any degree k, ${\displaystyle 0\leq k\leq D}$, ${\displaystyle \psi (x)=\psi _{i_{1}....i_{k}}(x)dx^{1}\wedge ...\wedge dx^{k}\in \Omega ^{(k)}(x)}$, where ${\displaystyle \Omega ^{(k)}(x)}$ is the space of all k-forms at point x. According to the example above, the temporal evolution of k-forms is given by, $${\displaystyle |\psi (t)\rangle ={\hat {M}}(\xi )_{t't}^{*}|\psi (t')\rangle ,}$$ where ${\displaystyle |\psi \rangle \in \Omega (X)=\bigoplus \nolimits _{k=0}^{D}\Omega ^{(k)}(X)}$ is a time-dependent "wavefunction", adopting the terminology of quantum theory. Unlike, say, trajectories in ${\displaystyle X}$, pullbacks are linear objects even for nonlinear ${\displaystyle X}$. As a linear object, the pullback can be averaged over the noise configurations, $${\displaystyle {\hat {\mathcal {M}}}_{tt'}=\iint {\hat {M}}(\xi )_{t't}^{*}P_{\text{noise}}(\xi ){\mathcal {D}}\xi \equiv \langle {\hat {M}}_{t't}(\xi )\rangle _{\text{noise}},}$$ where ${\displaystyle P_{\text{noise}}(\xi )}$ is the normalized probability functional of the noise and ${\displaystyle {\mathcal {D}}\xi }$ is the differential of the functional integration over the noise configurations. This is the generalized transfer operator (GTO) ^{[19] [20]} -- the dynamical systems theory counterpart of the stochastic evolution operator of the theory of SDEs and/or the Parisi-Sourlas approach.

Its explicit form can be derived by utilizing the concept of the chronological ordering of operators, $${\displaystyle {\hat {M}}(\xi )_{t't}^{*}\equiv {\mathcal {T}}e^{-\int _{t'}^{t}d\tau {\hat {L}}_{{\mathcal {F}}(\xi (\tau ))}}={\hat {1}}_{\Omega (X)}-\int _{t'}^{t}d\tau {\hat {L}}_{{\mathcal {F}}(\xi (\tau ))}+\int _{t'}^{t}d\tau _{1}\int _{t'}^{\tau _{1}}d\tau _{2}{\hat {L}}_{{\mathcal {F}}(\xi (\tau _{1}))}{\hat {L}}_{{\mathcal {F}}(\xi (\tau _{2}))}\dots ,}$$ which is the solution of ${\displaystyle \partial _{t}{\hat {M}}(\xi )_{t't}^{*}=-{\hat {L}}_{{\mathcal {F}}(\xi (t))}{\hat {M}}(\xi )_{t't}^{*},\left.M(\xi )_{t't}^{*}\right|_{t=t'}={\hat {1}}_{\Omega (X)},}$ where ${\displaystyle {\hat {L}}_{\mathcal {F}}=[{\hat {d}},{\hat {\imath }}_{\mathcal {F}}]}$ is the infinitesimal pullback or Lie derivative, expressed through Cartan formula, with ${\displaystyle {\hat {d}}}$ and ${\displaystyle {\hat {\imath }}_{\mathcal {F}}}$ being the exterior derivative and interior multiplication, respectively. Assuming Gaussian white noise, ${\displaystyle \langle \xi ^{a}(t)\rangle _{\text{noise}}=0,\langle \xi ^{a}(t)\xi ^{b}(t')\rangle _{\text{noise}}=\delta ^{ab}\delta (t-t')}$, and using, ${\displaystyle {\hat {L}}_{{\mathcal {F}}(\xi (t))}={\hat {L}}_{F}+\xi ^{a}(t)(2\Theta )^{1/2}{\hat {L}}_{G_{a}}}$, $${\displaystyle {\hat {\mathcal {M}}}_{tt'}=\langle {\mathcal {T}}e^{-\int _{t'}^{t}d\tau {\hat {L}}_{{\mathcal {F}}(\xi (\tau ))}}\rangle _{\text{noise}}={\hat {1}}_{\Omega (X)}-(t-t'){\hat {L}}_{F}+(t-t')^{2}{\hat {L}}_{F}^{2}/2+(t-t')\Theta {\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}}+...=e^{-(t-t'){\hat {H}}},}$$ where the infinitesimal GTO is given by $${\displaystyle {\hat {H}}={\hat {L}}_{F}-\Theta {\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}}=[{\hat {d}},{\hat {\bar {d}}}],{\text{ with }}{\hat {\bar {d}}}={\hat {\imath }}_{\mathcal {F}}-\Theta {\hat {\imath }}_{G_{a}}{\hat {L}}_{G_{a}}.}$$ From the point of view of the theory of SDEs, this GTO is a stochastic evolution operator (SEO) in Stratonovich interpretation. However, unlike SEOs in the theory of SDEs and/or the Parisi-Sourlas approach, the GTO has a clear-cut mathematical meaning, making it unique and eliminating the need for an additional interpretation beyond its definition.

Any pullback by a diffeomorphism commutes with ${\displaystyle {\hat {d}}}$ and the same holds for the GTO. In physical terms, this indicates the presence of a symmetry or, more precisely, a supersymmetry due to the nilpotency of the exterior derivative: ${\displaystyle {\hat {d}}^{2}=0}$. This supersymmetry is referred to as topological supersymmetry (TS), as the exterior derivative plays a fundamental role in algebraic topology.

Symmetries suggest degeneracy of eigenstates of evolution operators. In case of TS, if ${\displaystyle |\alpha \rangle }$ is an eigenstate of ${\displaystyle {\hat {H}}}$, then ${\displaystyle |\alpha '\rangle ={\hat {d}}|\alpha \rangle }$ is also an eigenstate with the same eigenvalue, provided that ${\displaystyle |\alpha '\rangle \neq 0}$.

The GTO is a pseudo-Hermitian operator.^[16] It has a complete bi-orthogonal eigensystem with the left and right eigenvectors, or the bras and the kets, related nontrivially. The eigensystems of GTO have a certain set of universal properties that limit the possible spectra of the physically meaningful models -- the ones with discrete spectra and with real parts of eigenvalues limited from below -- to the three major types presented in the figure on the right.^[33] These properties include:

• The eigenvalues are either real or come in complex conjugate pairs called in dynamical systems theory Reulle-Pollicott resonances. Each Reulle-Pollicott resonance can be thought of as a representation of the pseudo-time reversal (${\displaystyle \eta T}$-) symmetry.
• The GTO does not mix differential forms of different degrees: ${\displaystyle {\hat {H}}=diag({\hat {H}}^{(0)}...{\hat {H}}^{(D)})}$. In physics terms, the number of fermions is conserved. Each eigenstate has a well-defined degree.
• ${\displaystyle {\hat {H}}^{(0,D)}}$ do not break TS: ${\displaystyle {\text{min Re}}(\operatorname {spec} {\hat {H}}^{(0,D)})=0}$.
• Each De Rham cohomology class of the phase space provides one zero-eigenvalue supersymmetric "singlet" such that ${\displaystyle {\hat {d}}|\theta \rangle =0,\langle \theta |{\hat {d}}=0}$. Supersymmetric singlet from ${\displaystyle {\hat {H}}^{(D)}}$ is the stationary probability distribution. It is known in dynamical systems theory as "ergodic zero".
• All the other eigenstates are supersymmetric "doublets": ${\displaystyle {\hat {H}}|\alpha \rangle =H_{\alpha }|\alpha \rangle ,\;{\hat {H}}|\alpha '\rangle =H_{\alpha }|\alpha '\rangle }$ and ${\displaystyle \langle \alpha |{\hat {H}}=\langle \alpha |H_{\alpha },\langle \alpha '|{\hat {H}}=\langle \alpha '|H_{\alpha }}$, where ${\displaystyle H_{\alpha }}$ is the corresponding eigenvalue. The bras and kets are related via the supersymmetry operator ${\displaystyle |\alpha '\rangle ={\hat {d}}|\alpha \rangle ,\;\langle \alpha |=\langle \alpha '|{\hat {d}}}$.

From the point of view of the dynamical systems theory, a system can be characterized as chaotic if the spectral radius of the finite-time GTO -- referred to as the "pressure" -- is larger than unity. Under this condition, the partition function, $${\displaystyle Z_{tt'}=Tr{\hat {\mathcal {M}}}_{tt'}=\sum \nolimits _{\alpha }e^{-(t-t')H_{\alpha }},}$$ grows exponentially in the limit of infinitely long evolution signaling the exponential growth of the number of closed solutions -- the hallmark of chaotic dynamics. In terms of the infinitesimal GTO, this condition reads, $${\displaystyle \Delta =-\min _{\alpha }{\text{Re }}H_{\alpha }>0,}$$ where ${\displaystyle \Delta }$ is the rate of the exponential growth which can be recognized as a member of the family of dynamical entropies such as topological entropy. Spectra b and c in the figure satisfy this condition.

One notable advantage of defining stochastic chaos in this way, compared to other possible approaches, is its equivalence to the spontaneous breakdown of topological supersymmetry (see below). Consequently, through the Goldstone theorem, it has the potential to explain the experimental signature of chaotic behavior, commonly known as 1/f noise.

Another object of interest is the sharp trace of the GTO, $${\displaystyle W=Tr(-1)^{\hat {k}}{\hat {\mathcal {M}}}_{tt'}=\sum \nolimits _{\alpha }(-1)^{k_{\alpha }}e^{-(t-t')H_{\alpha }},}$$ where ${\displaystyle {\hat {k}}|\psi _{\alpha }\rangle =k_{\alpha }|\psi _{\alpha }\rangle }$ with ${\displaystyle {\hat {k}}}$ being the operator of the degree of the differential form. This is a fundamental object of topological nature known in physics as the Witten index. From the properties of the eigensystem of GTO, only supersymmetric singlets contribute to the Witten index, ${\displaystyle W=\sum \nolimits _{k=0}^{D}(-1)^{k}B_{k}=Eu.Ch(X)}$, where ${\displaystyle Eu.Ch.}$ is the Euler characteristic and B 's are Betti numbers that equal the number of supersymmetric singlets of the corresponding degree.

The Parisi–Sourlas method is the idea to use the following path integral representation of SDE: $${\displaystyle W=\left\langle \iint J(\xi )\left(\prod \nolimits _{\tau }\delta ^{D}({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))\right){\mathcal {D}}x\right\rangle _{\text{noise}}=\iint _{p.b.c.}e^{(Q,\Psi (\Phi ))}{\mathcal {D}}\Phi ,}$$ where ${\displaystyle \textstyle J(\xi )=\iint e^{-i\int d\tau d\tau '{\bar {\chi }}_{i}(\tau )J_{j}^{i}(\tau ,\tau ')\chi ^{j}(\tau ')}{\mathcal {D}}\chi {\mathcal {D}}{\bar {\chi }}}$ is the Jacobian of ${\displaystyle J_{j}^{i}(\tau ,\tau ')=\delta ({\dot {x}}^{i}(\tau )-{\mathcal {F}}^{i}(x(\tau ),\xi (\tau )))/\delta x^{j}(\tau ')}$ given as a functional integral over additional Grassmann fields ${\displaystyle \chi ,{\bar {\chi }}}$. In the second equality, yet another additional field called Lagrange multiplier, ${\displaystyle B}$, is introduced to "exponentiate" the ${\displaystyle \delta }$-functional limiting the functional integration only to solutions of SDE: $${\displaystyle \prod \nolimits _{\tau }\delta ^{D}({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))=\iint e^{i\int d\tau B(\tau )({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))}{\mathcal {D}}B.}$$ The notation ${\displaystyle \Phi }$ represents the set of all fields, ${\displaystyle xB\chi {\bar {\chi }}}$. The functional integration is over all closed paths and periodic boundary conditions (p.b.c.) are assumed for all the fields. The noise is assumed Gaussian white for simplicity. ${\displaystyle \textstyle \Psi =\int _{t'}^{t}d\tau (\imath _{\dot {x}}-{\bar {d}})(\tau )}$ is the so-called gauge fermion with ${\displaystyle \textstyle \imath _{\dot {x}}=i{\bar {\chi }}_{j}{\dot {x}}^{j}}$ and ${\textstyle \textstyle {\bar {d}}=\textstyle \imath _{F}-\Theta \imath _{G_{a}}L_{G_{a}},L_{G_{a}}=(Q,\imath _{G_{a}})}$. The topological supersymmetry can be defined as ${\displaystyle (Q,A(\Phi ))=\textstyle \int _{t'}^{t}d\tau (\chi ^{i}(\tau )\delta /\delta x^{i}(\tau )+B_{i}(\tau )\delta /\delta {\bar {\chi }}_{i}(\tau ))A(\Phi )}$, where ${\displaystyle A(\Phi )}$ as an arbitrary functional.

The Pasiri-Sourlas method can be understood as a BRST gauge fixing procedure. In this formalism, the Q-exact pieces like the action of the Parisi-Sourlas approach serve as gauge fixing tools. A common way to explain the BRST procedure is to say that the BRST symmetry generates the fermionic version of the gauge transformations, whereas its overall effect on the path integral is to limit the integration only to configurations that satisfy a specified gauge condition. This interpretation also applies to Parisi–Sourlas approach with the deformations of the trajectory and SDE playing the roles of the gauge transformations and the gauge condition respectively. This gauge fixing limits path integration only to solutions of SDE. Different solutions at a fixed noise can be understood as Gribov copies and the fermions of the theory can be identified as Faddeev–Popov ghosts.

The Parisi-Sourlas method is peculiar in that sense that it looks like gauge fixing of an empty theory -- the gauge fixing term is the only part of the action. This is a definitive feature of Witten-type topological field theories. Therefore, the Parisi-Sourlas method is a TFT ^{[24][23][25][27][28][29]} and as a TFT is has got objects that are topological invariants. The Parisi-Sourlas functional is one of them. It is essentially a pathintegral representation of the Witten index. The topological character of ${\displaystyle W}$ is seen by noting that the gauge-fixing character of the functional ensures that only solutions of the SDE contribute. Each solution provides either positive or negative unity: $${\displaystyle W=\textstyle \left\langle I_{N}(\xi )\right\rangle _{\text{noise}},{\text{ with }}I_{N}(\xi )=\sum _{\text{solutions}}\operatorname {sign} J(\xi ),}$$ being the index of the so-called Nicolai map, the map from the space of closed paths to the noise configurations making these closed paths solutions of the SDE, ${\textstyle \xi ^{a}(x)=G_{i}^{a}({\dot {x}}^{i}-F^{i})/(2\Theta )^{1/2}}$. The index of the map can be viewed as a realization of Poincaré–Hopf theorem on the infinite-dimensional space of close paths with the SDE playing the role of the vector field and with the solutions of the SDE playing the role of the critical points with index ${\displaystyle \operatorname {sign} J(\xi )=\operatorname {sign} {\text{Det }}\delta \xi /\delta x.}$ ${\textstyle I_{N}(\xi )}$ is a topological object independent of the noise configuration. It equals its own stochastic average which, in turn, equals the Witten index.

There are other classes of topological objects in TFTs including matrix elements on instantons. In fact, cohomological TFTs are often called intersection theory on instantons. From the STS viewpoint, instantons refers to quanta of transient dynamics, such as neuronal avalanches or solar flares, and complex or composite instantons represent nonlinear dynamical processes that occur in response to quenches -- external changes in parameters -- such as paper crumpling, protein folding etc. The application of the TFT aspect of STS to instantons remains largely unexplored.

The Parisi-Sourlas path integral with open boundary conditions is the stochastic evolution operator (SEO). Using the explicit form of the action ${\displaystyle (Q,\Psi (\Phi ))=\int _{t'}^{t}d\tau (iB{\dot {x}}+i{\dot {\chi }}{\bar {\chi }}-H)}$, where ${\displaystyle H=(Q,{\bar {d}})}$, the operator representation of the SEO can be derived as $${\displaystyle \iint _{{x\chi (t')=x_{i}\chi _{i}} \atop {x\chi (t)=x_{f}\chi _{f}}}e^{\int _{t'}^{t}d\tau (iB{\dot {x}}+i{\dot {\chi }}{\bar {\chi }}-H)}{\mathcal {D}}\Phi =\langle x_{f}\chi _{f}|e^{-(t-t'){\hat {H}}}|x_{i}\chi _{i}\rangle ,}$$ where the infinitesimal SEO ${\displaystyle {\hat {H}}=\left.H(xB\chi {\bar {\chi }})\right|_{B,{\bar {\chi }}\to {\hat {B}},{\hat {\bar {\chi }}}}}$, with ${\displaystyle i{\hat {B}}_{i}=\partial /\partial x^{i},i{\hat {\bar {\chi }}}_{i}=\partial /\partial \chi ^{i}}$, acts on ${\displaystyle x_{f},\chi _{f}}$. The explicit form of the SEO contains an ambiguity arising from the non-commutativity of momentum and position operators: ${\displaystyle Bx}$ in the path integral representation admits an entire ${\displaystyle \alpha }$-family of interpretations in the operator representation: ${\displaystyle \alpha {\hat {B}}{\hat {x}}+(1-\alpha ){\hat {x}}{\hat {B}}.}$

By definition, path integrals represent the continuous-time limit of a discrete-time evolution framework, which is equivalent to the traditional understanding of stochastic dynamics in the theory of SDEs -- the continuous-time limit of stochastic difference equations. Consequently, the same ambiguity arises in the theory of SDEs, where different choices of ${\displaystyle \alpha }$ are referred to as different interpretations of SDEs with ${\displaystyle \alpha =1,1/2,0}$ being respectively the Ito, Stratonovich, and Kolmogorov interpretations.

This intrinsic ambiguity can only be removed by imposing some additional conditions or principles. In quantum theory, the condition is the requirement for a Hermitian Hamiltonian, which is satisfied by the Weyl symmetrization rule corresponding to ${\displaystyle \alpha =1/2}$. In STS, the condition is that the SEO must equal the GTO, which is also achieved at ${\displaystyle \alpha =1/2}$. Therefore, only the Stratonovich interpretation of SDEs is consistent with the dynamical systems theory approach. Other interpretations differ only by the shifted flow vector field in the corresponding SEO, ${\displaystyle F_{\alpha }=F-\Theta (2\alpha -1)(G_{a}\cdot \partial )G_{a}}$, which does not introduce, however, any new mathematics beyond the that of the Stratonovich interpretation of SDEs. At the same time, other interpretations are important in the context of discrete-time stochastic evolution and numerical implementation of SDEs.

The wavefunctions in STS depend not only on the original variables of the SDE but also on their supersymmetric partners ${\displaystyle \chi }$. These Grassmann numbers, or fermions, represent the differentials of the differential forms in the dynamical systems theory interpetation of STS.^[28] The fermions are intrinsically linked to stochastic Lyapunov exponents, ^[18] This suggests, particularly, that under conditions of spontaneous TS breaking, the effective theory for these fermions -- referred to as goldstinos in this context -- is essentially a theory of the butterfly effect.

The response of the model can be analyzed using the concept of generating functional: $${\displaystyle G(\eta )=-\log \lim _{T\to \infty }\langle g|{\hat {M}}_{T/2,-T/2}(\eta )|g\rangle ,}$$ where ${\displaystyle \eta }$ denotes external probing fields, ${\displaystyle {\hat {M}}_{T/2,-T/2}(\eta )}$ is the perturbed SEO/GTO, and ${\displaystyle |g\rangle }$ is the ground state. The ground state must be selected from the eigenstates with the smallest real part of the eigenvalue to ensure the stability of the model's response, ${\displaystyle {\text{Re }}H_{g}=\min \nolimits _{\alpha }{\text{Re }}H_{\alpha }.}$

The functional dependence of the generating functional on the probing fields describes how the ground state reacts to external perturbations. Under conditions of spontaneously broken TS, there exists another eigenstate with the same eigenvalue, ${\displaystyle H_{g}}$. In line with the Goldstone theorem, this degeneracy of the ground state implies the presence of a gapless excitation that must mediate long-range response. This mechanism qualitatively explains the widespread occurrence of long-range behavior in chaotic dynamics known as 1/f noise. However, a more rigorous theoretical explanation of 1/f noise remains an open problem.

When ${\displaystyle H_{g}}$ is complex, pseudo-time-reversal symmetry is also spontaneously broken. In the context of kinematic dynamo, this situation corresponds to the overall rotation of the galactic magnetic field ^[33]. The implications of complex ${\displaystyle H_{g}}$ in a more general setting remain unexplored.

STS predicts the existence of a distinct phase at the border of chaos where the dynamics is an endless sequence of noise-induced instantons, as seen in earthquakes, solar flares, neuronal avalanches etc. Unlike deterministic models, where the spontaneous breakdown of TS results from the non-integrability of the flow vector field, the presence of noise necessitates a phase in which TS is broken by the condensation of noise-induced instantons. In the deterministic limit, these instantons disappear, causing this phase to collapse onto the boundary of conventional deterministic chaos. Therefore, this phase can be referred to as noise-induced chaos. This offers a potential theoretical explanation for self-organized criticality -- a phenomenological approach based on the idea that certain SDEs exhibit a mysterious tendency to fine-tune themselves to a phase transition between order and chaos. ^{[34] [35]}

• Stochastic quantization