# 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, 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. By describing chaos as a spontaneous breakdown of topological supersymmetry, STS aims to explain long-range dynamical phenomena such as 1/f noise and the power-law statistics of instantonic processes like earthquakes and neuroavalanches.

## Overview

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 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 exhibits a stochastic variant of the butterfly effect. 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.

## History and relation to other theories

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.
In parallel, the concept of the generalized transfer operator have been introduced in the dynamical systems theory. 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.
The Parisi-Sourlas method has been recognized as a member of Witten-type or cohomological topological field theory, a class of models to which STS also belongs.

## Dynamical systems theory perspective

A continuous-time non-autonomous dynamical system can be defined as, ${\dot {x}}(t)=F(x(t))+(2\Theta )^{1/2}G_{a}(x(t))\xi ^{a}(t)\equiv {\mathcal {F}}(\xi (t)),$ where x∈X{\textstyle x\in X} is a point in the phase space which can be assumed to be a closed smooth manifold, $F(x)\in TX_{x}$ is a sufficiently smooth flow vector field from the tangent space of $X$, and $G_{a}\in TX,a=1,\ldots ,D,D=\dim X$ is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, $\xi (t)\in \mathbb {R} ^{D}$, which is called additive/multiplicative depending on whether $G_{a}$'s are independent/dependent on the position on $X$. 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, ${\mathcal {F}}$.
Even for non-differentiable $\xi (t)$, the solutions/trajectories of the ODE are differentiable with respect to initial conditions. In other words, there exists a two-parameter family of noise-configuration-dependent diffeomorphisms: $M(\xi )_{tt'}:X\to X,M(\xi )_{tt'}\circ M(\xi )_{t't''}=M(\xi )_{tt''},\left.M(\xi )_{tt'}\right|_{t=t'}={\text{Id}}_{X},$ such that the solution of the ODE with initial condition $x(t')=x'$ can be expressed as $x(t)=M(\xi )_{tt'}(x')$.
The dynamics can now be defined as follows: if at time $t'$, the system is described by the probability distribution $P(x)$, then the average value of some function $f:X\to \mathbb {R}$ at a later time $t$ is given by: ${\bar {f}}(t)=\int _{X}f\left(M(\xi )_{tt'}(x)\right)P(x)dx^{1}\wedge ...\wedge dx^{D}=\int _{X}f(x){\hat {M}}(\xi )_{t't}^{*}\left(P(x)dx^{1}\wedge ...\wedge dx^{D}\right).$ Here ${\hat {M}}(\xi )_{t't}^{*}$ is action or pullback induced by the inverse map, $M(\xi )_{tt'}^{-1}=M(\xi )_{t't}$, 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 other possible degrees k, $0\leq k\leq D$, $\psi (x)=\psi _{i_{1}....i_{k}}(x)dx^{1}\wedge ...\wedge dx^{k}\in \Omega ^{(k)}(x)$, where $\Omega ^{(k)}(x)$ is the space all k-forms at point x. According to the example above, the temporal evolution of k-forms is given by, $|\psi (t)\rangle ={\hat {M}}(\xi )_{t't}^{*}|\psi (t')\rangle ,$ where $|\psi \rangle \in \Omega (X)=\bigoplus \nolimits _{k=0}^{D}\Omega ^{(k)}(X)$ is a time-dependent "wavefunction", adopting the terminology of quantum theory. Unlike, say, trajectories in $X$, pullbacks are linear objects even for nonlinear $X$. As a linear object, the pullback can be averaged over the noise configurations, ${\hat {\mathcal {M}}}_{tt'}=\langle {\hat {M}}_{t't}(\xi )\rangle _{\text{noise}},$ This is the generalized transfer operator (GTO)
-- the dynamical systems theory counterpart of the stochastic evolution operator of the theory of SDEs and/or the Parisi-Sourlas approach. For Gaussian white noise, $\langle \xi ^{a}(t)\rangle _{\text{noise}}=0,\langle \xi ^{a}(t)\xi ^{b}(t')\rangle _{\text{noise}}=\delta ^{ab}\delta (t-t')$, ${\hat {\mathcal {M}}}_{tt'}=e^{-(t-t'){\hat {H}}},$ where the infinitesimal GTO is given by ${\hat {H}}={\hat {L}}_{F}-\Theta {\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}}=[{\hat {d}},{\hat {\bar {d}}}],$ where ${\hat {\bar {d}}}={\hat {\imath }}_{\mathcal {F}}-\Theta {\hat {\imath }}_{G_{a}}{\hat {L}}_{G_{a}},$ and ${\hat {L}}_{Z}=[{\hat {d}},{\hat {\imath }}_{Z}]$ is the infinitesimal pullback or Lie derivative, expressed through Cartan formula, with ${\hat {d}}$ and ${\hat {\imath }}_{Z}$ being, respectively, the exterior derivative and interior multiplication along arbitrary vector field $Z$.
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 ${\hat {d}}$ 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: ${\hat {d}}^{2}=0$. 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 $|\alpha \rangle$ is an eigenstate of ${\hat {H}}$, then $|\alpha '\rangle ={\hat {d}}|\alpha \rangle$ is also an eigenstate with the same eigenvalue, provided that $|\alpha '\rangle \neq 0$.

### GTO eigensystem

The GTO is a pseudo-Hermitian operator. 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. These properties include: the eigenvalues are either real or come in complex conjugate pairs called in dynamical systems theory Reulle-Pollicott resonances; each eigenstate has a well-defined degree; ${\hat {H}}^{(0,D)}$ do not break TS, ${\text{min Re}}(\operatorname {spec} {\hat {H}}^{(0,D)})=0$; each De Rham cohomology provides one zero-eigenvalue supersymmetric "singlet" such that ${\hat {d}}|\theta \rangle =0,\langle \theta |{\hat {d}}=0$ and the singlet from ${\hat {H}}^{(D)}$ is the stationary probability distribution known as "ergodic zero"; all the other eigenstates are non-supersymmetric "doublets" related by TS: ${\hat {H}}|\alpha \rangle =H_{\alpha }|\alpha \rangle ,\;{\hat {H}}|\alpha '\rangle =H_{\alpha }|\alpha '\rangle$ and $\langle \alpha |{\hat {H}}=\langle \alpha |H_{\alpha },\langle \alpha '|{\hat {H}}=\langle \alpha '|H_{\alpha }$, where $H_{\alpha }$ is the corresponding eigenvalue, and $|\alpha '\rangle ={\hat {d}}|\alpha \rangle ,\;\langle \alpha |=\langle \alpha '|{\hat {d}}$.

### Stochastic generalization of chaos

In dynamical systems theory, a system can be characterized as chaotic if the spectral radius of the finite-time GTO is larger than unity. Under this condition, the partition function,
$Z_{tt'}=Tr{\hat {\mathcal {M}}}_{tt'}=\sum \nolimits _{\alpha }e^{-(t-t')H_{\alpha }},$ 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, $\Delta =-\min _{\alpha }{\text{Re }}H_{\alpha }>0,$ where $\Delta$ is the rate of the exponential growth which is known as "pressure", 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.

### Sharp trace and Witten Index

Another object of interest is the sharp trace of the GTO, $W=Tr(-1)^{\hat {k}}{\hat {\mathcal {M}}}_{tt'}=\sum \nolimits _{\alpha }(-1)^{k_{\alpha }}e^{-(t-t')H_{\alpha }},$ where ${\hat {k}}|\psi _{\alpha }\rangle =k_{\alpha }|\psi _{\alpha }\rangle$ with ${\hat {k}}$ 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, $W=\sum \nolimits _{k=0}^{D}(-1)^{k}B_{k}=Eu.Ch(X)$, where $Eu.Ch.$ is the Euler characteristic and B 's are Betti numbers that equal the number of supersymmetric singlets of the corresponding degree.

## Physical Perspective
### Parisi–Sourlas method and gauge-fixing

The idea of the Parisi–Sourlas method is to rewrite the partition function of the noise in terms of the dynamical variables of the model using BRST gauge-fixing procedure. The resulting expression is the Witten index, whose physical meaning is (up to a topological factor) the partition function of the noise.
As the first step toward the pathintegral representation of the Witten index, the pathintegration over dynamical variables are formally introduced into the expression of the partition function of the noise: $1=\langle 1\rangle _{\text{noise}}=\iint P(\xi ){\mathcal {D}}\xi \to \iint _{p.b.c}P(\xi ){\mathcal {D}}\xi {\mathcal {D}}x,$ where the noise is assumed Gaussian white for simplicity with normalized probability functional, $P(\xi )\propto e^{-\int d\tau \xi ^{2}/2}$ and the functional integration over dynamical variables goes over closed paths or paths with periodic boundary conditions (p.b.c). This expression can be viewed as a redundant theory of the noise. Its "action" is independent of the dynamical variables. This independence can be interpreted as a local symmetry of the model with respect to all possible deformations of the paths. This local symmetry can be gauge-fixed using the SDE as a gauge condition, which leads to the following representations of the Witten index: $W=\iint _{p.b.c}{\mathcal {D}}\xi {\mathcal {D}}xP(\xi )J(\xi )\left(\prod \nolimits _{\tau }\delta ^{D}({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))\right),$ where the $\delta$-functional limits the integration only to solutions of SDE, which can be understood in the context of BRST gauge fixing as Gribov copies, and $\textstyle J(\xi )$ is the Jacobian of $J_{j}^{i}(\tau ,\tau ')=\delta ({\dot {x}}^{i}(\tau )-{\mathcal {F}}^{i}(x(\tau ),\xi (\tau )))/\delta x^{j}(\tau ')$ that compensates (up to a sign) the Jacobian from the $\delta$-functional. With the help of the Faddeev–Popov ghosts $\chi ,{\bar {\chi }}$ and the Lagrange multiplier, $B$, such that $\textstyle J(\xi )=\iint e^{-i\int d\tau d\tau '{\bar {\chi }}_{i}(\tau )J_{j}^{i}(\tau ,\tau ')\chi ^{j}(\tau ')}{\mathcal {D}}\chi {\mathcal {D}}{\bar {\chi }}$ and $\prod \nolimits _{\tau }\delta ^{D}({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))=\iint e^{i\int d\tau B(\tau )({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))}{\mathcal {D}}B$: $W=\iint _{p.b.c.}{\mathcal {D}}\xi P(\xi )e^{(Q,\Psi (\xi ,\Phi ))}{\mathcal {D}}\Phi =\iint _{p.b.c.}e^{(Q,\Psi (\Phi ))}{\mathcal {D}}\Phi ,$ where $\Phi$ represents all the fields, $xB\chi {\bar {\chi }}$, $\Psi (\xi ,\Phi )=\int d\tau \imath _{{\dot {x}}(\tau )-{\mathcal {F}}(\tau )}$, with $\textstyle \imath _{\dot {x}}=i{\bar {\chi }}_{j}{\dot {x}}^{j}$ being the pathintegral version of the interior multiplication, and the BRST supersymmetry can be defined as $Q=\textstyle \int d\tau (\chi ^{i}(\tau )\delta /\delta x^{i}(\tau )+B_{i}(\tau )\delta /\delta {\bar {\chi }}_{i}(\tau ))$. In the second equality sign, the noise is integrated out and $\textstyle \Psi =\int _{t'}^{t}d\tau (\imath _{\dot {x}}-{\bar {d}})(\tau )$ is the gauge fermion with d$^¯$=ı$_F$−Θı$_{Ga}$L$_{Ga}$, and L$_{Ga}$=(Q, ı$_{Ga}$){\textstyle \textstyle {\bar {d}}=\textstyle \imath _{F}-\Theta \imath _{G_{a}}L_{G_{a}}, {\text{ and }}L_{G_{a}}=(Q, \imath _{G_{a}})} being the pathintegral version of Lie derivative.

### STS as a topological field theory

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 and as a TFT it 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 $W$ 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: $W=\langle \iint _{p.b.c}{\mathcal {D}}xJ(\xi )\left(\prod \nolimits _{\tau }\delta ^{D}({\dot {x}}(\tau )-{\mathcal {F}}(x(\tau ),\xi (\tau )))\right)\rangle _{\text{noise}}=\textstyle \left\langle I_{N}(\xi )\right\rangle _{\text{noise}},{\text{ with }}I_{N}(\xi )=\sum _{\text{solutions}}\operatorname {sign} J(\xi ),$ 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, ξ$^a$(x)=G$_i$$^a$(x$^˙$$^i$−F$^i$)/(2Θ)$^{1/2}${\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 $\operatorname {sign} J(\xi )=\operatorname {sign} {\text{Det }}\delta \xi /\delta x.$ I$_N$(ξ){\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.

### Operator representation and Ito-Stratonovich dilemma

Just like the partition function of the noise that it represents, the Witten index contains no information about the system's dynamics and cannot be used directly to investigate the dynamics in the system. The information on the dynamics is contained in the stochastic evolution operator (SEO) -- the Parisi-Sourlas path integral with open boundary conditions. Using the explicit form of the action $(Q,\Psi (\Phi ))=\int _{t'}^{t}d\tau (iB{\dot {x}}+i{\dot {\chi }}{\bar {\chi }}-H)$, where $H=(Q,{\bar {d}})$, the operator representation of the SEO can be derived as $\iint _{{x\chi (t')=x_{i}\chi _{i}} \atop {x\chi (t)=x_{f}\chi _{f}}}e^{\int _{t'}^{t}d\tau (iB{\dot {x}}+i{\dot {\chi }}{\bar {\chi }}-H)}{\mathcal {D}}\Phi =\langle x_{f}\chi _{f}|e^{-(t-t'){\hat {H}}}|x_{i}\chi _{i}\rangle ,$ where the infinitesimal SEO ${\hat {H}}=\left.H(xB\chi {\bar {\chi }})\right|_{B,{\bar {\chi }}\to {\hat {B}},{\hat {\bar {\chi }}}}$, with $i{\hat {B}}_{i}=\partial /\partial x^{i},i{\hat {\bar {\chi }}}_{i}=\partial /\partial \chi ^{i}$. The explicit form of the SEO contains an ambiguity arising from the non-commutativity of momentum and position operators: $Bx$ in the path integral representation admits an entire $\alpha$-family of interpretations in the operator representation: $\alpha {\hat {B}}{\hat {x}}+(1-\alpha ){\hat {x}}{\hat {B}}.$
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 $\alpha$ are referred to as different interpretations of SDEs with $\alpha =1,1/2,0$ 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 $\alpha =1/2$. In STS, the condition is that the SEO must equal the GTO, which is also achieved at $\alpha =1/2$. 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, $F_{\alpha }=F-\Theta (2\alpha -1)(G_{a}\cdot \partial )G_{a}$, which does not introduce, however, any new mathematics beyond that of the Stratonovich interpretation. At the same time, other interpretations are important in the context of discrete-time stochastic evolution and numerical implementation of SDEs.

### Meaning of wavefunctions and the butterfly effect

The wavefunctions in STS depend not only on the original variables of the SDE but also on their supersymmetric partners $\chi$. These Grassmann numbers, or fermions, represent the differentials of the differential forms in the dynamical systems theory interpretation of STS. The fermions are intrinsically linked to stochastic Lyapunov exponents that determine butterfly effect. As a result, 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.

## 1/f noise and topological supersymmetry breaking

The response of the model can be analyzed using the concept of generating functional: $G(\eta )=-\log \lim _{T\to \infty }\langle g|{\hat {M}}_{T/2,-T/2}(\eta )|g\rangle ,$ where $\eta$ denotes external probing fields, ${\hat {M}}_{T/2,-T/2}(\eta )$ is the perturbed SEO/GTO, and $|g\rangle$ 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, ${\text{Re }}H_{g}=\min \nolimits _{\alpha }{\text{Re }}H_{\alpha }.$
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, $H_{g}$. 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 picture qualitatively explains the widespread occurrence of long-range behavior in chaotic dynamics known as 1/f noise. A more rigorous theoretical explanation of 1/f noise remains an open problem.

### Pseudo-time reversal symmetry breaking

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

## Self-organized criticality and noise-induced chaos

Since the late 80's, the concept of the Edge of chaos has emerged -- a finite-width phase at the boundary of conventional chaos, where dynamics is often dominated by power-law distributed instantonic processes such as solar flares, earthquakes, and neuronal avalanches.
This phase has also been recognized as potentially significant for information processing. Its phenomenological understanding is largely based on the concepts of self-adaptation and self-organization.
STS offers the following explanation for the Edge of chaos (see figure on the right). In the presence of noise, the TS can be spontaneously broken not only by the non-integrability of the flow vector field, as in deterministic chaos, but also by noise-induced instantons.
Under this condition, the dynamics must be dominated by instantons with power-law distributions, as dictated by the Goldstone theorem. In the deterministic limit, the noise-induced instantons vanish, causing the phase hosting this type of noise-induced dynamics to collapse onto the boundary of the deterministic chaos.