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'''Supersymmetric theory of stochastic dynamics''' ('''STS''') is a multidisciplinary approach to [[Stochastic process|stochastic]] dynamics on the intersection of [[dynamical systems theory]],
[[statistical physics]],
[[stochastic differential equation]]s (SDE),
[[Topological quantum field theories|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 theory|chaos]] as a [[Spontaneous symmetry breaking|spontaneous breakdown]] of [[Supersymmetry#Supersymmetry in dynamical systems|topological supersymmetry]], STS aims to explain long-range dynamical phenomena such as [[Pink noise|1/f]], [[Flicker noise|flicker]], and [[crackling noise|crackling]] noises and the [[power law|power-law statistics]] of instantonic processes like earthquakes and neuroavalanches.

== Overview ==

The traditional approach to stochastic dynamics focuses on the [[Fokker–Planck_equation|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",<ref name=":15"/> whose evolution determines [[Lyapunov exponent]]s.<ref name=":Graham"/> 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]]<ref name=":6"/> and [[dynamical systems theory]] defines their dynamics by the generalized transfer operator (GTO)<ref name=":0"/><ref name=":19"/> -- 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 [[Topological space|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 [[pink noise|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,<ref name=":9">{{Cite journal|last1=Parisi|first1=G.|last2=Sourlas|first2=N.|date=1979|title=Random Magnetic Fields, Supersymmetry, and Negative Dimensions|journal=Physical Review Letters|volume=43|issue=11|pages=744–745|doi=10.1103/PhysRevLett.43.744|bibcode=1979PhRvL..43..744P}}</ref><ref name=":15">{{Cite journal|last=Parisi|first=G.|title=Supersymmetric field theories and stochastic differential equations|journal=Nuclear Physics B|language=en|volume=206|issue=2|pages=321–332|doi=10.1016/0550-3213(82)90538-7|year=1982|bibcode=1982NuPhB.206..321P}}</ref> 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 quantization|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]].<ref name=Aharony1976>{{cite journal|author=Aharony, A.|author2=Imry, Y.|author3=Ma, S.K.|year=1976|title=Lowering of dimensionality in phase transitions with random fields|journal=Physical Review Letters|volume=37|issue=20|pages=1364–1367|doi=10.1103/PhysRevLett.37.1364|bibcode=1976PhRvL..37.1364A }}</ref> 
Since then, the so-emerged supersymmetry of Langevin SDEs has been addressed from a few different angles <ref>{{Cite journal|last1=Cecotti|first1=S|last2=Girardello|first2=L|date=1983-01-01|title=Stochastic and parastochastic aspects of supersymmetric functional measures: A new non-perturbative approach to supersymmetry|journal=Annals of Physics|volume=145|issue=1|pages=81–99|doi=10.1016/0003-4916(83)90172-0|bibcode=1983AnPhy.145...81C|doi-access=free}}</ref><ref>{{Cite journal|last=Zinn-Justin|first=J.|date=1986-09-29|title=Renormalization and stochastic quantization|journal=Nuclear Physics B|volume=275|issue=1|pages=135–159|doi=10.1016/0550-3213(86)90592-4|bibcode=1986NuPhB.275..135Z}}</ref><ref>{{Cite journal|last1=Dijkgraaf|first1=R.|last2=Orlando|first2=D.|last3=Reffert|first3=S.|date=2010-01-11|title=Relating field theories via stochastic quantization|journal=Nuclear Physics B|volume=824|issue=3|pages=365–386|doi=10.1016/j.nuclphysb.2009.07.018|bibcode=2010NuPhB.824..365D|arxiv=0903.0732|s2cid=2033425}}</ref><ref name=":12">{{Cite journal|last=Kurchan|first=J.|date=1992-07-01|title=Supersymmetry in spin glass dynamics|journal=Journal de Physique I|language=en|volume=2|issue=7|pages=1333–1352|doi=10.1051/jp1:1992214|issn=1155-4304|bibcode=1992JPhy1...2.1333K|s2cid=124073976|url=https://hal.science/jpa-00246625/document }}</ref><ref name=":14" /> including the [[Fluctuation-dissipation theorem|fluctuation dissipation theorems]],<ref name=":12" /> [[Jarzynski equality]],<ref name=":20">{{cite arXiv|last1=Mallick|first1=K.|last2=Moshe|first2=M.|last3=Orland|first3=H.|date=2007-11-13|title=Supersymmetry and Nonequilibrium Work Relations|eprint=0711.2059|class=cond-mat.stat-mech}}</ref> [[Onsager reciprocal relations|Onsager principle of microscopic reversibility]],<ref name=":11">{{Cite journal|last=Gozzi|first=E.|date=1984|title=Onsager principle of microscopic reversibility and supersymmetry|journal=Physical Review D|volume=30|issue=6|pages=1218–1227|doi=10.1103/physrevd.30.1218|bibcode=1984PhRvD..30.1218G}}</ref> solutions of Fokker–Planck equations,<ref>{{Cite journal|last=Bernstein|first=M.|date=1984|title=Supersymmetry and the Bistable Fokker-Planck Equation|journal=Physical Review Letters|volume=52|issue=22|pages=1933–1935|doi=10.1103/physrevlett.52.1933|bibcode=1984PhRvL..52.1933B}}</ref> [[self-organization]],<ref>{{Cite journal|last1=Olemskoi|first1=A. I|last2=Khomenko|first2=A. V|last3=Olemskoi|first3=D. A|date=2004-02-01|title=Field theory of self-organization|journal=Physica A: Statistical Mechanics and Its Applications|volume=332|pages=185–206|doi=10.1016/j.physa.2003.10.035|bibcode=2004PhyA..332..185O|url=http://essuir.sumdu.edu.ua/handle/123456789/16485}}</ref> etc.

The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including [[classical mechanics]],<ref name=":1">{{Cite journal|last1=Gozzi|first1=E.|last2=Reuter|first2=M.|title=Classical mechanics as a topological field theory|journal=Physics Letters B|language=en|volume=240|issue=1–2|pages=137–144|doi=10.1016/0370-2693(90)90422-3|year=1990|bibcode=1990PhLB..240..137G|url=https://cds.cern.ch/record/204132}}</ref><ref name=":16">{{Cite journal|last=Niemi|first=A. J.|title=A lower bound for the number of periodic classical trajectories|journal=Physics Letters B|language=en|volume=355|issue=3–4|pages=501–506|doi=10.1016/0370-2693(95)00780-o|year=1995|bibcode=1995PhLB..355..501N}}</ref> its stochastic generalization,<ref name=":13">{{Cite journal|last1=Tailleur|first1=J.|last2=Tănase-Nicola|first2=S.|last3=Kurchan|first3=J.|date=2006-02-01|title=Kramers Equation and Supersymmetry|journal=Journal of Statistical Physics|language=en|volume=122|issue=4|pages=557–595|doi=10.1007/s10955-005-8059-x|issn=0022-4715|bibcode=2006JSP...122..557T|arxiv=cond-mat/0503545|s2cid=119716999}}</ref> and higher-order Langevin SDEs.<ref name=":14">{{Cite journal|last1=Kleinert|first1=H.|last2=Shabanov|first2=S. V.|date=1997-10-27|title=Supersymmetry in stochastic processes with higher-order time derivatives|journal=Physics Letters A|volume=235|issue=2|pages=105–112|doi=10.1016/s0375-9601(97)00660-9|bibcode=1997PhLA..235..105K|arxiv=quant-ph/9705042|s2cid=119459346}}</ref>
The theory of pseudo-Hermitian supersymmetric operators
<ref name=":2">{{Cite journal|last=Mostafazadeh|first=A.|date=2002-07-19|title=Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries|journal=Journal of Mathematical Physics|volume=43|issue=8|pages=3944–3951|doi=10.1063/1.1489072|issn=0022-2488|bibcode=2002JMP....43.3944M|arxiv=math-ph/0203005|s2cid=7096321}}</ref> 
and the relation between the Parisi-Sourlas method and [[Lyapunov exponent]]s <ref name=":Graham">{{Cite journal|last=Graham|first=R.|date=1988|title=Lyapunov Exponents and Supersymmetry of Stochastic Dynamical Systems|journal=EPL|language=en|volume=5|issue=2|pages=101–106|doi=10.1209/0295-5075/5/2/002|issn=0295-5075|bibcode=1988EL......5..101G|s2cid=250788554 }}</ref>
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.<ref name=":10">{{Cite journal|last=Ovchinnikov|first=I. V.|date=2016-03-28|title=Introduction to Supersymmetric Theory of Stochastics|journal=Entropy|language=en|volume=18|issue=4|pages=108|doi=10.3390/e18040108|bibcode=2016Entrp..18..108O|arxiv=1511.03393|s2cid=2388285|doi-access=free}}</ref>

In parallel, the concept of the generalized [[transfer operator]] have been introduced in the [[dynamical systems theory]].<ref name=":0">{{cite journal|date=2002|title=Dynamical Zeta Functions and Transfer Operators|url=http://www.ams.org/notices/200208/fea-ruelle.pdf|journal=Notices of the AMS|volume=49|issue=8|pages=887|author=Reulle, D.}}</ref><ref name=":19">{{Cite journal|last=Ruelle|first=D.|date=1990-12-01|title=An extension of the theory of Fredholm determinants|journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques|language=en|volume=72|issue=1|pages=175–193|doi=10.1007/bf02699133|s2cid=121869096|issn=0073-8301|url=http://www.numdam.org/item/PMIHES_1990__72__175_0/}}</ref> 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.<ref>{{Cite book|title=Stochastic differential geometry at Saint-Flour|last1=Ancona|first1=A.|last2=Elworthy|first2=K. D.|last3=Emery|first3=M.|last4=Kunita|first4=H.|date=2013|publisher=Springer|isbn=9783642341700|oclc=811000422}}</ref><ref>{{Cite book|title=Stochastic flows and stochastic differential equations|last=Kunita|first=H.|date=1997|publisher=Cambridge University Press|isbn=978-0521599252|oclc=36864963}}</ref>

The Parisi-Sourlas method has been recognized <ref name=":Baulieu_Grossman"/><ref name=":1"/> as a member of Witten-type or cohomological [[topological quantum field theory|topological field theory]],<ref name=":3">{{Cite journal|last1=Birmingham|first1=D|last2=Blau|first2=M.|last3=Rakowski|first3=M.|last4=Thompson|first4=G.|title=Topological field theory|journal=Physics Reports|language=en|volume=209|issue=4–5|pages=129–340|doi=10.1016/0370-1573(91)90117-5|year=1991|bibcode=1991PhR...209..129B|url=https://cds.cern.ch/record/218572}}
</ref><ref name=":4">{{Cite journal|last=Witten|first=E.|date=1988-09-01|title=Topological sigma models|journal=Communications in Mathematical Physics|language=en|volume=118|issue=3|pages=411–449|doi=10.1007/BF01466725|issn=0010-3616|bibcode=1988CMaPh.118..411W|s2cid=34042140|url=http://projecteuclid.org/euclid.cmp/1104162092}}</ref><ref>{{Cite journal|last1=Baulieu|first1=L.|last2=Singer|first2=I.M.|journal=Communications in Mathematical Physics|language=en|issue=2|pages=227–237|volume=125|date=1988|title=The topological sigma model|doi=10.1007/BF01217907|s2cid=120150962}}</ref><ref name=":5">{{Cite journal|last=Witten|first=E.|date=1988-09-01|title=Topological quantum field theory|journal=Communications in Mathematical Physics|language=en|volume=117|issue=3|pages=353–386|doi=10.1007/BF01223371|issn=0010-3616|bibcode=1988CMaPh.117..353W|s2cid=43230714|url=http://projecteuclid.org/euclid.cmp/1104161738}}</ref><ref name=":6">{{Cite journal|last=Witten|first=E.|date=1982|title=Supersymmetry and Morse theory|journal=Journal of Differential Geometry|language=EN|volume=17|issue=4|pages=661–692|doi=10.4310/jdg/1214437492|issn=0022-040X|doi-access=free}}</ref><ref name=":7">{{Cite journal|last=Labastida|first=J. M. F.|date=1989-12-01|title=Morse theory interpretation of topological quantum field theories|journal=Communications in Mathematical Physics|language=en|volume=123|issue=4|pages=641–658|doi=10.1007/BF01218589|issn=0010-3616|bibcode=1989CMaPh.123..641L|citeseerx=10.1.1.509.3123|s2cid=53555484}}</ref><ref>{{Cite journal|last=Nicolai|first=H.|date=1980-12-22|title=Supersymmetry and functional integration measures|journal=Nuclear Physics B|volume=176|issue=2|pages=419–428|doi=10.1016/0550-3213(80)90460-5|bibcode=1980NuPhB.176..419N|url=https://cds.cern.ch/record/134045|hdl=11858/00-001M-0000-0013-5E89-E|hdl-access=free}}</ref><ref>{{Cite journal|last=Nicolai|first=H.|date=1980-01-28|title=On a new characterization of scalar supersymmetric theories|journal=Physics Letters B|volume=89|issue=3|pages=341–346|doi=10.1016/0370-2693(80)90138-0|bibcode=1980PhLB...89..341N|url=https://cds.cern.ch/record/133513/files/197911274.pdf}}</ref> a class of models to which STS also belongs.
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== Dynamical systems theory perspective ==
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A continuous-time non-autonomous dynamical system can be defined as,
<math display="block"> \dot x(t) = F(x(t))+(2\Theta)^{1/2}G_a(x(t))\xi^a(t)\equiv{\mathcal F}(\xi(t)),</math>
where <math display="inline">x\in X </math> is a point in the phase space which can be assumed to be a [[Closed manifold|closed]] [[smooth manifold]], <math> F(x)\in TX_x </math> is a sufficiently smooth flow [[vector field]] from the [[tangent space]] of <math> X</math>, and  <math> G_a \in TX, a=1, \ldots, D, D=\dim X </math> is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, <math>\xi(t)\in\mathbb{R}^D</math>, which is called [[Additive noise|additive]]/[[Multiplicative noise|multiplicative]] depending on whether <math> G_a </math>'s are independent/dependent on the position on <math>X</math>. 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, <math>\mathcal F</math>.

Even for non-differentiable <math>\xi(t)</math>, the solutions/trajectories of the ODE are differentiable with respect to initial conditions.<ref>{{Cite journal|last=Slavík|first=A.|title=Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters|journal=Journal of Mathematical Analysis and Applications|language=en|volume=402|issue=1|pages=261–274|doi=10.1016/j.jmaa.2013.01.027|year=2013|doi-access=free}}</ref> In other words, there exists a two-parameter family of noise-configuration-dependent [[diffeomorphism]]s:
<math display="block">M(\xi)_{tt'}:X\to X, M(\xi)_{tt'}\circ M(\xi)_{t't''}=M(\xi)_{tt''}, \left.M(\xi)_{ t t'}\right|_{t=t'} = \text{Id}_X,</math> 
such that the solution of the ODE with initial condition <math>x(t')=x'</math> can be expressed as <math>x(t) = M(\xi)_{tt'}(x')</math>. 
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The dynamics can now be defined as follows: if at time <math>t'</math>, the system is described by the probability distribution <math>P(x)</math>, then the average value of some function <math>f:X\to\mathbb{R}</math> at a later time <math> t </math> is given by:
<math display="block">\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). </math>
Here <math> \hat M(\xi)^*_{t't} </math> is action or [[Pullback (differential geometry)#Pullback of differential forms|pullback]] induced by the ''inverse'' map, <math>M(\xi)_{tt'}^{-1}=M(\xi)_{t't}</math>, 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, <math>0\le k\le D</math>, <math>\psi(x) = \psi_{i_1....i_k}(x)dx^1\wedge ... \wedge dx^k\in\Omega^{(k)}(x)</math>, where <math>\Omega^{(k)}(x)</math> is the space all k-forms at point x. 
According to the example above, the temporal evolution of k-forms is given by,
<math display="block">|\psi(t)\rangle =   \hat M(\xi)_{t't}^*|\psi(t')\rangle,</math>
where <math>|\psi\rangle\in\Omega(X)=\bigoplus\nolimits_{k=0}^D\Omega^{(k)}(X)</math> is a time-dependent "wavefunction", adopting the terminology of quantum theory. Unlike, say, trajectories in <math>X</math>, pullbacks are linear objects even for nonlinear <math>X</math>. As a linear object, the pullback can be averaged over the noise configurations,
<math display="block" > \hat{\mathcal M }_{tt'} = \langle\hat M_{t't}(\xi)\rangle_{\text{noise}},</math>
This is the generalized transfer operator (GTO)
<ref name=":0" />
<ref name=":19" /> 
-- 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, <math> \langle \xi^a(t) \rangle_{\text{noise}} =0, \langle\xi^a(t)\xi^b(t')\rangle_{\text{noise}} = \delta^{ab}\delta(t-t')</math>,
<math display="block" > \hat{\mathcal M }_{tt'} = e^{-(t-t')\hat H}, </math>
where the ''infinitesimal'' GTO is given by
<math display="block" > \hat H = \hat L_F - \Theta \hat L_{G_a}\hat L_{G_a} = [\hat d, \hat {\bar d}],</math> 
where <math>\hat {\bar d} = \hat{\imath}_{\mathcal{F}} - \Theta \hat{\imath}_{G_a}\hat L_{G_a},</math> and <math> \hat L_{Z} = [\hat d, \hat{\imath}_{Z}]</math> is the infinitesimal pullback or [[Lie derivative]], expressed through [[Cartan formula]], with <math>\hat d</math> and <math>\hat{\imath}_{Z}</math> being, respectively, the [[exterior derivative]] and [[interior multiplication]] along arbitrary vector field <math>Z</math>. 

From the point of view of the theory of SDEs, this GTO is a stochastic evolution operator (SEO) in [[Stratonovich integral|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 <math>\hat d</math> 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]]: <math>\hat d^2=0</math>. 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 <math>|\alpha\rangle</math> is an eigenstate of <math>\hat H</math>, then <math>|\alpha'\rangle = \hat d|\alpha\rangle</math> is also an eigenstate with the same eigenvalue, provided that <math>|\alpha'\rangle \ne 0</math>.  

=== GTO eigensystem ===

[[File:Possible types of spectra of SEO of STS 3 sphere.png|thumb|The three possible types of SEO spectra on a 3D sphere. Each line of triples of graphs represents <math>spec {\hat H}^{(n)},</math> <math>n=0,...3</math>. for the three types of spectra. Black dots at the origin for the first and the last lines represent supersymmetric eigenstates from the zeroth and the third cohomologies of the 3 sphere.  For types '''b''' and '''c''', the (fastest growing) ground (eigen)states are non-supersymmetric because they have non-zero eigenvalues. TS is broken spontaneously. Vertical arrowed lines visualize supersymmetry operator.]]

The GTO is a pseudo-Hermitian operator.<ref name=":2" /> 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.<ref name=":22"/> 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 (<math>\eta T</math>-) symmetry.
* The GTO does not mix differential forms of different degrees: <math> \hat H = diag (\hat H^{(0)}...\hat H^{(D)}) </math>. In physics terms, the number of fermions is conserved. Each eigenstate has a well-defined degree.
* <math> \hat H^{(0,D)} </math> do not break TS: <math> \text{min Re} (\operatorname{spec} \hat H^{(0,D)}) = 0 </math>.
* Each De Rham cohomology class of the phase space provides one supersymmetric "singlet" such that <math> \hat d |\theta\rangle = 0, \langle \theta | \hat d = 0 </math>.
* All supersymmetric singlets have zero eigenvalue because <math>\hat H</math> is <math>\hat d</math>-exact, i.e., has the form of a bi-graded commutator with <math>\hat d</math>. Supersymmetric singlet from <math> \hat H^{(D)} </math> is the stationary probability distribution. It is known in dynamical systems theory as "ergodic zero". 
* All the other eigenstates are supersymmetric "doublets": <math> \hat H|\alpha\rangle = H_\alpha |\alpha\rangle,\; \hat H|\alpha'\rangle = H_\alpha |\alpha'\rangle </math> and <math> \langle\alpha| \hat H= \langle \alpha| H_\alpha, \langle\alpha'| \hat H = \langle\alpha'| H_\alpha </math>, where <math> H_\alpha </math> is the corresponding eigenvalue. The bras and kets are related via the supersymmetry operator <math> |\alpha'\rangle = \hat d |\alpha\rangle,\;\langle \alpha | = \langle\alpha'|\hat d </math>.

=== 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,  
<math display="block" > Z_{tt'} = Tr \hat{\mathcal M }_{tt'} = \sum\nolimits_{\alpha}e^{-(t-t')H_\alpha}, </math>
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, 
<math display="block" > \Delta = - \min_\alpha \text{Re }H_\alpha > 0, </math>
where <math > \Delta </math> 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 [[pink noise|1/f noise]].

=== Sharp trace and Witten Index ===

Another object of interest is the ''sharp'' trace of the GTO,
<math display="block" > W = Tr (-1)^{\hat k} \hat{\mathcal M }_{tt'} = \sum\nolimits_\alpha (-1)^{k_\alpha}e^{-(t-t')H_\alpha}, </math>
where <math> \hat k |\psi_\alpha\rangle = k_\alpha |\psi_\alpha\rangle</math> with <math>\hat k</math> 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, <math>W=\sum\nolimits_{k=0}^D (-1)^k B_k=Eu.Ch(X)</math>, where <math>Eu.Ch.</math> is the [[Euler characteristic]] and ''B'' 's are [[Betti numbers]] that equal the number of supersymmetric singlets of the corresponding degree.
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== Physical Perspective ==
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=== Parisi–Sourlas method and gauge-fixing ===<!-- EDIT BELOW THIS LINE -->
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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 quantization|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: 
<math display="block">
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,
</math>
where the noise is assumed Gaussian white for simplicity with normalized probability functional, <math>P(\xi)\propto e^{-\int d\tau \xi^2/2}</math> 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:
<math display="block">
W=\iint_{p.b.c} {\mathcal D}\xi {\mathcal D}x P(\xi) J(\xi) \left(\prod\nolimits_\tau \delta ^D (\dot x(\tau) - {\mathcal F}(x(\tau),\xi(\tau)))\right),
</math>
where the  <math>\delta</math>-functional limits the integration only to solutions of SDE, which can be understood in the context of BRST gauge fixing as [[Gribov copies]], and <math> \textstyle J(\xi) </math> is the Jacobian of <math>J^i_j(\tau,\tau') = \delta(\dot x^i (\tau) - 
{\mathcal F}^i (x(\tau),\xi(\tau)))/ \delta x^j (\tau')</math> that compensates (up to a sign) the Jacobian from the  <math>\delta</math>-functional. With the help of the [[Faddeev–Popov_ghost|Faddeev–Popov ghosts]] <math> \chi, \bar\chi</math> and the Lagrange multiplier, <math> B</math>, such that <math>  \textstyle J(\xi)=
\iint e^{-i\int d\tau d\tau' \bar\chi_i(\tau)J^i_j(\tau,\tau')\chi^j(\tau')} {\mathcal D}\chi{\mathcal D}\bar\chi
</math> and <math> \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 </math>:
<math display="block">
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,
</math>
where <math> \Phi </math> represents all the fields, <math>xB\chi\bar\chi</math>, <math>
\Psi(\xi,\Phi) = \int d\tau \imath_{\dot x(\tau) - \mathcal F(\tau)} </math>, with <math> \textstyle \imath_{\dot x} = i \bar\chi_j\dot x^j </math> being the pathintegral version of the interior multiplication, and the BRST supersymmetry can be defined as <math> Q = \textstyle \int d\tau(\chi^i(\tau)\delta/\delta x^i(\tau) + B_i(\tau)\delta/\delta \bar\chi_i(\tau)) </math>. In the second equality sign, the noise is integrated out and <math>\textstyle \Psi = \int_{t'}^td\tau (\imath_{\dot x} - \bar d)(\tau)</math> is the gauge fermion with <math display="inline"> \textstyle \bar d = \textstyle \imath_F - \Theta \imath_{G_a} L_{G_a}, \text{ and } L_{G_a}=(Q,\imath_{G_a})</math> 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 [[topological quantum field theory|Witten-type topological field theories]]. Therefore, the Parisi-Sourlas method is a TFT <ref name=":3" /><ref name=":Baulieu_Grossman">{{Cite journal|last1=Baulieu|first1=L.|last2=Grossman|first2=B.|date=1988|journal=Physics Letters B|title=A topological interpretation of stochastic quantization|language=en|volume=212|issue=3|pages=351–356|doi=10.1016/0370-2693(88)91328-7|bibcode=1988PhLB..212..351B}}</ref><ref name=":4" /><ref name=":5" /><ref name=":6" /><ref name=":7" /> 
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 <math>W</math> 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: 
<math display="block">W = 
\langle \iint_{p.b.c} {\mathcal D}x J(\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),</math>
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, <math display="inline"> \xi^a(x) = G^a_i(\dot x^i - F^i)/(2\Theta)^{1/2} </math>. 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 <math>
\operatorname{sign}J(\xi) = \operatorname{sign}\text{Det }\delta \xi/\delta x.
</math> 
<math display="inline"> I_N(\xi) </math> is a topological object independent of the noise configuration. It equals its own stochastic average which, in turn, equals the [[#Sharp trace and Witten index|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 <math>(Q,\Psi(\Phi))=\int_{t'}^t d\tau (iB\dot x + i\dot \chi {\bar \chi} - H)</math>, where <math>H=(Q,\bar d)</math>, the operator representation of the SEO can be derived as
<math display="block">
\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,
</math>
where the infinitesimal SEO <math>\hat H = \left.H(xB\chi\bar\chi)\right|_{B,\bar\chi\to\hat B,{\hat {\bar\chi}}}</math>, with <math>i\hat B_i=\partial/\partial x^i, i\hat{\bar\chi}_i=\partial/\partial\chi^i</math>. The explicit form of the SEO contains an ambiguity arising from the non-commutativity of momentum and position operators: <math> Bx </math> in the path integral representation admits an entire <math>\alpha</math>-family of interpretations in the operator representation: <math> \alpha \hat B \hat x + (1-\alpha)\hat x \hat B.</math>

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 equation|stochastic difference equations]]. Consequently, the same ambiguity arises in the theory of SDEs, where different choices of <math>\alpha</math> are referred to as different interpretations of SDEs with <math> \alpha=1,1/2,0 </math> being respectively the [[Itô calculus|Ito]], [[Stratonovich integral|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 [[Wigner–Weyl transform|Weyl symmetrization]] rule corresponding to <math>\alpha=1/2</math>. In STS, the condition is that the SEO must equal the GTO, which is also achieved at <math>\alpha=1/2</math>. Therefore, only the Stratonovich interpretation of SDEs is consistent with the [[#Dynamical systems theory perspective|dynamical systems theory approach]]. Other interpretations differ only by the shifted flow vector field in the corresponding SEO, <math> F_\alpha = F - \Theta(2\alpha-1)(G_a\cdot\partial) G_a </math>, 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.

[[File:STS Phase Diagram.png|thumb|Stochastic dynamical systems can be classified based on whether TS is spontaneously broken or unbroken (ordered or symmetric), and whether the flow vector field is integrable or non-integrable, i.e., chaotic. The symmetric phase with unbroken TS is denoted as (T). The ordered non-integrable phase can be referred to as chaos (C), as it hosts conventional deterministic chaos. The ordered integrable phase is called noise-induced chaos (N), as the dynamics is dominated by noise-induced instantons, which disappear in the deterministic limit so that N-phase collapses onto the border of deterministic chaos. As the noise intensity increases, TS is eventually restored]]

=== 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 <math>\chi</math>. These Grassmann numbers, or fermions, represent the differentials of the differential forms in the dynamical systems theory interpretation of STS.<ref name=":6"/> The fermions are intrinsically linked to stochastic Lyapunov exponents<ref name=":Graham"/> 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 function (physics)|generating functional]]:
<math display="block"> G(\eta) = -\log \lim_{T\to\infty} \langle g | \hat M _{T/2, -T/2}(\eta) | g \rangle, </math>
where <math>\eta</math> denotes external probing fields, <math> \hat M _{T/2, -T/2}(\eta)</math> is the perturbed SEO/GTO, and <math> | g \rangle </math> 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, <math>\text{Re } H_g=\min\nolimits_{\alpha}\text{Re }H_\alpha.</math>

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, <math>H_g</math>. 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 [[pink noise|1/f noise]]. A more rigorous theoretical explanation of 1/f noise remains an open problem.

=== Pseudo-time reversal symmetry breaking ===

When <math>H_g</math> is complex, [[#GTO eigensystem|pseudo-time-reversal symmetry]] is also spontaneously broken. In the context of [[Dynamo theory#Kinematic dynamo theory|kinematic dynamo]], this situation corresponds to rotation of the galactic magnetic field.<ref name=":22">{{Cite journal|last1=Ovchinnikov|first1=I.V.|last2=Ensslin|first2=T. A.|date=2016|title=Kinematic dynamo, supersymmetry breaking, and chaos|journal=Physical Review D|volume=93|issue=8|pages=085023|doi=10.1103/PhysRevD.93.085023|bibcode=2016PhRvD..93h5023O|arxiv=1512.01651|s2cid=59367815}}</ref> The implications of complex <math>H_g</math> in a more general setting remain unexplored.

== Self-organized criticality and noise-induced chaos ==

Since the late 80's,<ref name=EOC-T-30>{{cite book|last=A. Bass|first=Thomas|title = The Predictors : How a Band of Maverick Physicists Used Chaos Theory to Trade Their Way to a Fortune on Wall Street|url =https://books.google.com/books?id=MQ-xGC7BdS0C&pg=PA138|publisher = Henry Holt and Company |year =1999|isbn =9780805057560 |page =[https://books.google.com/books?id=MQ-xGC7BdS0C&pg=PA138 138] |access-date=12 November 2020}}</ref><ref name=EOC-T-20>{{cite web|last=H. Packard|first=Norman|title = Adaptation Toward the Edge of Chaos|url =https://books.google.com/books?id=8prgtgAACAAJ|publisher = University of Illinois at Urbana-Champaign, Center for Complex Systems Research |year =1988|access-date=12 November 2020}}</ref> 
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.
<ref>{{cite book|first=Markus|last=Aschwanden|title=Self-Organized Criticality in Astrophysics|year=2011|publisher =Springer}}</ref>
This phase has also been recognized as potentially significant for information processing.<ref name=EOC-T-17>{{cite journal|last1=Langton|first1=Christopher.|title=Studying artificial life with cellular automata|journal=Physica D|date=1986|volume=22|issue=1–3|pages=120–149|doi=10.1016/0167-2789(86)90237-X|bibcode=1986PhyD...22..120L |hdl=2027.42/26022|hdl-access=free}}</ref><ref name=EOC-T-28>{{cite web|last2=Young|first2=Karl|last1=P. Crutchfleld|first1=James|title=Computation at the Onset of Chaos|url=http://csc.ucdavis.edu/~cmg/papers/CompOnset.pdf|year=1990|access-date=11 November 2020}}</ref>
Its phenomenological understanding is largely based on the concepts of [[Adaptive system|self-adaptation]] and [[self-organized criticality|self-organization]].<ref>{{Cite journal|last1=Watkins|first1=N. W.|last2=Pruessner|first2=G.|last3=Chapman|first3=S. C.|last4=Crosby|first4=N. B.|last5=Jensen|first5=H. J.|date=2016-01-01|title=25 Years of Self-organized Criticality: Concepts and Controversies|journal=Space Science Reviews|language=en|volume=198|issue=1–4|pages=3–44|doi=10.1007/s11214-015-0155-x|issn=0038-6308|bibcode=2016SSRv..198....3W|arxiv=1504.04991|s2cid=34782655}}</ref><ref>{{Cite journal|last1=Bak|first1=P.|last2=Tang|first2=C.|last3=Wiesenfeld|first3=K.|date=1987|title=Self-organized criticality: An explanation of the 1/f noise|journal=Physical Review Letters|volume=59|issue=4|pages=381–384|doi=10.1103/PhysRevLett.59.381|pmid=10035754|bibcode=1987PhRvL..59..381B|s2cid=7674321 }}</ref>

STS offers the following explanation for the [[Edge of chaos]] (see figure on the right).{{cn|date=February 2025}} In the presence of noise, the TS can be spontaneously broken not only by the [[Integrable system|non-integrability]] of the flow vector field, as in deterministic chaos, but also by noise-induced instantons.
<ref> {{cite journal|last1=Witten|first1=Edward|title=Dynamical breaking of supersymmetry|journal=Nuclear Physics B|date=1988|volume=188|issue=3|pages=513-554|doi=10.1016/0550-3213(81)90006-7}} </ref>
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.

== See also ==

* [[Stochastic quantization]]
* [[Supersymmetric quantum mechanics]]
* [[Topological quantum field theory]]

== References ==
{{Reflist}}

{{Industrial and applied mathematics}}

<!--- Categories --->
[[Category:Supersymmetry]]
[[Category:Chaos theory]]
[[Category:Mathematical physics]]
[[Category:Applied and interdisciplinary physics]]
[[Category:Complex systems theory]]
[[Category:Self-organization]]
[[Category:Stochastic processes]]