Supersymmetric theory of stochastic dynamics#Spontaneous supersymmetry breaking and chaos

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/state 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.

The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by Giorgio Parisi and Nicolas Sourlas,{{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}}{{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}} 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, {{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 }} the so-emerged supersymmetry of Langevin SDEs has since been addressed from a few different angles {{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}}{{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}}{{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}}{{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 }} including the fluctuation-dissipation theorems, Jarzynski equality,{{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}} Onsager principle of microscopic reversibility,{{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}} solutions of Fokker–Planck equations,{{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}} self-organization,{{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}} etc.

The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including classical mechanics,{{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}}{{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}} its stochastic generalization,{{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}} and higher-order Langevin SDEs.{{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}}

The theory of pseudo-Hermitian supersymmetric operators

{{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}}

and the relation between the Parisi-Sourlas method and Lyapunov exponents {{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 }}

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.{{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}}

In parallel, the concept of the generalized transfer operator have been introduced in the dynamical systems theory.{{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.}}{{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/}} 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.{{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}}{{Cite book|title=Stochastic flows and stochastic differential equations|last=Kunita|first=H.|date=1997|publisher=Cambridge University Press|isbn=978-0521599252|oclc=36864963}}

The Parisi-Sourlas method has been recognized as a member of Witten-type or cohomological topological field theory,{{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}}

{{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}}{{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}}{{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}}{{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}}{{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}}{{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}}{{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}} a class of models to which STS also belongs.

The physicist's way to look at a stochastic differential equation is essentially a continuous-time non-autonomous dynamical system that can be defined as:

$$\backslash dot\; x(t)\; =\; F(x(t))+(2\backslash Theta)^\{1/2\}G\_a(x(t))\backslash xi^a(t)\backslash equiv\{\backslash mathcal\; F\}(\backslash xi(t)),$$

where $x\backslash in\; X$ is a point in a closed smooth manifold, $X$, called in dynamical systems theory a state space while in physics, where $X$ is often a symplectic manifold with half of variables having the meaning of momenta, it is called the phase space. Further, $F(x)\backslash in\; TX\_x$ is a sufficiently smooth flow vector field from the tangent space of $X$ having the meaning of deterministic law of evolution, and $G\_a\; \backslash in\; TX,\; a=1,\; \backslash ldots,\; D,\; D=\backslash dim\; X$ is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, $\backslash xi(t)\backslash in\backslash mathbb\{R\}^D$, 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, $\backslash mathcal\; F$. The solutions/trajectories of this ODE are differentiable with respect to initial conditions even for non-differentiable $\backslash xi(t)$'s.{{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}} In other words, there exists a two-parameter family of noise-configuration-dependent diffeomorphisms:

$$M(\backslash xi)\_\{tt\text{'}\}:X\backslash to\; X,\; M(\backslash xi)\_\{tt\text{'}\}\backslash circ\; M(\backslash xi)\_\{t\text{'}t$$}=M(\xi)_{tt}, \left.M(\xi)_{ t t'}\right|_{t=t'} = \text{Id}_X,

such that the solution of the ODE with initial condition $x(t\text{'})=x\text{'}$ can be expressed as $x(t)\; =\; M(\backslash xi)\_\{tt\text{'}\}(x\text{'})$.

The dynamics can now be defined as follows: if at time $t\text{'}$, the system is described by the probability distribution $P(x)$, then the average value of some function $f:X\backslash to\backslash mathbb\{R\}$ at a later time $t$ is given by:

$$\backslash bar\; f(t)\; =\; \backslash int\_X\; f\backslash left(M(\backslash xi)\_\{tt\text{'}\}(x)\backslash right)\; P(x)\; dx^1\; \backslash wedge\; ...\; \backslash wedge\; dx^D\; =\; \backslash int\_X\; f(x)\; \backslash hat\; M(\backslash xi)\_\{t\text{'}t\}^*\backslash left(P(x)\; dx^1\; \backslash wedge\; ...\; \backslash wedge\; dx^D\backslash right).$$

Here $\backslash hat\; M(\backslash xi)^*\_\{t\text{'}t\}$ is action or pullback induced by the inverse map, $M(\backslash xi)\_\{tt\text{'}\}^\{-1\}=M(\backslash xi)\_\{t\text{'}t\}$, 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, $0\backslash le\; k\backslash le\; D$, $\backslash psi(x)\; =\; \backslash psi\_\{i\_1....i\_k\}(x)dx^1\backslash wedge\; ...\; \backslash wedge\; dx^k\backslash in\backslash Omega^\{(k)\}(x)$, where $\backslash Omega^\{(k)\}(x)$ is the space all k-forms at point x.

According to the example above, the temporal evolution of k-forms is given by,

$$|\backslash psi(t)\backslash rangle\; =\; \backslash hat\; M(\backslash xi)\_\{t\text{'}t\}^*|\backslash psi(t\text{'})\backslash rangle,$$

where $|\backslash psi\backslash rangle\backslash in\backslash Omega(X)=\backslash bigoplus\backslash nolimits\_\{k=0\}^D\backslash Omega^\{(k)\}(X)$ 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 leading to 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.

With the help of the concept of Lie derivative, $\backslash hat\; L\_\{\{\backslash mathcal\; F\}(\backslash tau)\}\; =\; \backslash hat\; L\_\{F\}\; +(2\backslash Theta)^\{1/2\}\backslash xi^a(\backslash tau)\backslash hat\; L\_\{G\_a\}$, which is essentially the infinitesimal pullback satisfying, in particular, the following equation, $\backslash partial\_t\; \backslash hat\; M(\backslash xi)^*\_\{t\text{'}t\}\; =\; -\; \backslash hat\; L\_\{\{\backslash mathcal\; F\}(\backslash tau)\}\; \backslash hat\; M(\backslash xi)^*\_\{t\text{'}t\}$, which integrates to

$$\backslash hat\; M(\backslash xi)^*\_\{t\text{'}t\}\; =\; \backslash hat\; 1\; -\; \backslash int\_\{t\text{'}\}^t\; \backslash hat\; L\_\{\backslash mathcal\; F(\backslash tau)\}\; d\backslash tau\; +\; \backslash int\_\{t\text{'}\}^t\; \backslash hat\; L\_\{\backslash mathcal\; F(\backslash tau\_1)\}\; d\backslash tau\_1\backslash int\_\{t\text{'}\}^\{\backslash tau\_1\}\; \backslash hat\; L\_\{\backslash mathcal\; F(\backslash tau\_2)\}\; d\backslash tau\_2\; +...,$$

for the initial condition $\backslash hat\; M(\backslash xi)^*\_\{t\text{'}t\}|\_\{t=t\text{'}\}=\backslash hat\; 1$, and assuming Gaussian white noise, $\backslash langle\; \backslash xi^a(t)\; \backslash rangle\_\{\backslash text\{noise\}\}\; =0,\; \backslash langle\backslash xi^a(t)\backslash xi^b(t\text{'})\backslash rangle\_\{\backslash text\{noise\}\}\; =\; \backslash delta^\{ab\}\backslash delta(t-t\text{'})$..., the GTO can be derived as

$$\backslash hat\{\backslash mathcal\; M\; \}\_\{tt\text{'}\}\; =\; \backslash langle\; M(\backslash xi)^*\_\{t\text{'}t\}\; \backslash rangle\_\{\backslash text\{noise\}\}\; =\; \backslash hat\; 1\; -\; (t-t\text{'})\backslash hat\; H\; +\; ...\; =\; e^\{-(t-t\text{'})\backslash hat\; H\}.$$

Here, the infinitesimal GTO,

$$\backslash hat\; H\; =\; \backslash hat\; L\_F\; -\; \backslash Theta\; \backslash hat\; L\_\{G\_a\}\backslash hat\; L\_\{G\_a\}\; =\; [\backslash hat\; d,\; \backslash hat\; \{\backslash bar\; d\}],$$

and $\backslash hat\; \{\backslash bar\; d\}\; =\; \backslash hat\{\backslash imath\}\_\{\backslash mathcal\{F\}\}\; -\; \backslash Theta\; \backslash hat\{\backslash imath\}\_\{G\_a\}\backslash hat\; L\_\{G\_a\}$ which follows from Cartan formula for Lie derivative, e.g., $\backslash hat\; L\_F\; =\; [\backslash hat\; d,\; \backslash hat\{\backslash imath\}\_\{F\}]$ with the square brackets denoting bi-graded commutator and $\backslash hat\; d$ and $\backslash hat\{\backslash imath\}\_\{F\}$ being, respectively, the exterior derivative and interior multiplication, along with the nilpotency of the exterior differentiation suggesting, particularly, that $[\backslash hat\; d,\; \backslash hat\; A][\backslash hat\; d,\; \backslash hat\; B]\; =\; [\backslash hat\; d,\; \backslash hat\; A\; [\backslash hat\; d,\; \backslash hat\; B]]$.

Any pullback by a diffeomorphism commutes with $\backslash hat\; d$ 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: $\backslash hat\; d^2=0$. 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 $|\backslash alpha\backslash rangle$ is an eigenstate of $\backslash hat\; H$, then $|\backslash alpha\text{'}\backslash rangle\; =\; \backslash hat\; d|\backslash alpha\backslash rangle$ is also an eigenstate with the same eigenvalue, provided that $|\backslash alpha\text{'}\backslash rangle\; \backslash ne\; 0$.

File:Possible types of spectra of SEO of STS 3 sphere.png

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; $\backslash hat\; H^\{(0,D)\}$ do not break TS, $\backslash text\{min\; Re\}\; (\backslash operatorname\{spec\}\; \backslash hat\; H^\{(0,D)\})\; =\; 0$; each De Rham cohomology provides one zero-eigenvalue supersymmetric "singlet" such that $\backslash hat\; d\; |\backslash theta\backslash rangle\; =\; 0,\; \backslash langle\; \backslash theta\; |\; \backslash hat\; d\; =\; 0$ and the singlet from $\backslash hat\; H^\{(D)\}$ is the stationary probability distribution known as "ergodic zero"; all the other eigenstates are non-supersymmetric "doublets" related by TS: $\backslash hat\; H|\backslash alpha\backslash rangle\; =\; H\_\backslash alpha\; |\backslash alpha\backslash rangle,\backslash ;\; \backslash hat\; H|\backslash alpha\text{'}\backslash rangle\; =\; H\_\backslash alpha\; |\backslash alpha\text{'}\backslash rangle$ and $\backslash langle\backslash alpha|\; \backslash hat\; H=\; \backslash langle\; \backslash alpha|\; H\_\backslash alpha,\; \backslash langle\backslash alpha\text{'}|\; \backslash hat\; H\; =\; \backslash langle\backslash alpha\text{'}|\; H\_\backslash alpha$, where $H\_\backslash alpha$ is the corresponding eigenvalue, and $|\backslash alpha\text{'}\backslash rangle\; =\; \backslash hat\; d\; |\backslash alpha\backslash rangle,\backslash ;\backslash langle\; \backslash alpha\; |\; =\; \backslash langle\backslash alpha\text{'}|\backslash hat\; d$.

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,

$$Z\_\{tt\text{'}\}\; =\; Tr\; \backslash hat\{\backslash mathcal\; M\; \}\_\{tt\text{'}\}\; =\; \backslash sum\backslash nolimits\_\{\backslash alpha\}e^\{-(t-t\text{'})H\_\backslash alpha\},$$

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,

$$\backslash Delta\; =\; -\; \backslash min\_\backslash alpha\; \backslash text\{Re\; \}H\_\backslash alpha\; >\; 0,$$

where $\backslash Delta$ 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.

It is easily to see that the properties of GTO spectra imply that stochastic chaos is only possible when $\backslash text\{dim\; \}X\backslash ge3$. This can be viewed as a stochastic generalization of the Poincaré–Bendixson theorem.

Another object of interest is the sharp trace of the GTO,

$$W\; =\; Tr\; (-1)^\{\backslash hat\; k\}\; \backslash hat\{\backslash mathcal\; M\; \}\_\{tt\text{'}\}\; =\; \backslash sum\backslash nolimits\_\backslash alpha\; (-1)^\{k\_\backslash alpha\}e^\{-(t-t\text{'})H\_\backslash alpha\},$$

where $\backslash hat\; k\; |\backslash psi\_\backslash alpha\backslash rangle\; =\; k\_\backslash alpha\; |\backslash psi\_\backslash alpha\backslash rangle$ with $\backslash hat\; k$ 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, $W=\backslash sum\backslash nolimits\_\{k=0\}^D\; (-1)^k\; B\_k=Eu.Ch(X)$, where $Eu.Ch.$ is the Euler characteristic and B 's are Betti numbers that equal the numbers of supersymmetric singlets of the corresponding degree.

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:

\langle 1 \rangle_{\text{noise}} = \iint {\mathcal P}(\xi) {\mathcal D}\xi \to \iint_{p.b.c} {\mathcal D}x {\mathcal P}(\xi){\mathcal D}\xi,

where the noise is again assumed Gaussian white with the normalized probability functional, $\{\backslash mathcal\; P\}(\backslash xi)\backslash propto\; e^\{-\backslash int\; d\backslash tau\; (\backslash xi(\backslash tau))^2/2\},\; \backslash langle\; 1\; \backslash rangle\_\{\backslash text\{noise\}\}\; =\; 1,$ and the functional integration over dynamical variables goes over closed paths, i.e., paths with periodic boundary conditions (p.b.c). The expression in the r.h.s. 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 continuous deformations of the paths. This local symmetry can be gauge-fixed using the SDE as a gauge condition, which leads to the following representation of the Witten index:

W=\iint_{p.b.c} J(\xi) \left(\prod\nolimits_\tau \delta ^D (\dot x(\tau) - {\mathcal F}(x(\tau),\xi(\tau)))\right) {\mathcal D}x {\mathcal P}(\xi) {\mathcal D}\xi=\iint_{p.b.c.} e^{(Q,\Psi(\xi,\Phi)) }{\mathcal D}\Phi {\mathcal P}(\xi) {\mathcal D}\xi = \iint_{p.b.c.} e^{(Q,\Psi(\Phi))}{\mathcal D}\Phi,

where the $\backslash delta$-functional limits the integration only to solutions of SDE, which can be understood in this context as Gribov copies, $\backslash textstyle\; J(\backslash xi)$ is the Jacobian compensating (up to a sign) the Jacobian from the $\backslash delta$-functional, $\backslash Phi\; =xB\backslash chi\backslash bar\backslash chi$ is the collection of fields that includes, besides the original field $x$, the Faddeev–Popov ghosts $\backslash chi,\; \backslash bar\backslash chi$ and the Lagrange multiplier, $B$, and $$

\Psi(\xi,\Phi) = \int d\tau \imath_{\dot x(\tau) - \mathcal F(\tau)} , with $\backslash textstyle\; \backslash imath\_\{\backslash dot\; x\}\; =\; i\; \backslash bar\backslash chi\_j\backslash dot\; x^j$ being the pathintegral version of the interior multiplication, the topological and/or BRST supersymmetry is,

$$Q\; =\; \backslash textstyle\; \backslash int\; d\backslash tau(\backslash chi^i(\backslash tau)\backslash delta/\backslash delta\; x^i(\backslash tau)\; +\; B\_i(\backslash tau)\backslash delta/\backslash delta\; \backslash bar\backslash chi\_i(\backslash tau)),$$

and, in the last equality, the noise is integrated out and $\backslash textstyle\; \backslash Psi\; =\; \backslash int\_\{t\text{'}\}^td\backslash tau\; (\backslash imath\_\{\backslash dot\; x\}\; -\; \backslash bar\; d)(\backslash tau)$ is the gauge fermion with $\backslash textstyle\; \backslash bar\; d\; =\; \backslash textstyle\; \backslash imath\_F\; -\; \backslash Theta\; \backslash imath\_\{G\_a\}\; L\_\{G\_a\},\; \backslash text\{\; and\; \}\; L\_\{G\_a\}=(Q,\backslash imath\_\{G\_a\})$ being the pathintegral version of Lie derivative.

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 {{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}}

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}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),

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, $\backslash xi^a(x)\; =\; G^a\_i(\backslash dot\; x^i\; -\; F^i)/(2\backslash Theta)^\{1/2\}$. 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.

$I\_N(\backslash xi)$ 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.

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 $(Q,\backslash Psi(\backslash Phi))=\backslash int\_\{t\text{'}\}^t\; d\backslash tau\; (iB\backslash dot\; x\; +\; i\backslash dot\; \backslash chi\; \{\backslash bar\; \backslash chi\}\; -\; H)$, where $H=(Q,\backslash bar\; d)$, 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 $\backslash hat\; H\; =\; \backslash left.H(xB\backslash chi\backslash bar\backslash chi)\backslash right|\_\{B,\backslash bar\backslash chi\backslash to\backslash hat\; B,\{\backslash hat\; \{\backslash bar\backslash chi\}\}\}$, with $i\backslash hat\; B\_i=\backslash partial/\backslash partial\; x^i,\; i\backslash hat\{\backslash bar\backslash chi\}\_i=\backslash partial/\backslash partial\backslash chi^i$. 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 $\backslash alpha$-family of interpretations in the operator representation: $\backslash alpha\; \backslash hat\; B\; \backslash hat\; x\; +\; (1-\backslash alpha)\backslash hat\; x\; \backslash hat\; B.$ The same ambiguity arises in the theory of SDEs, where different choices of $\backslash alpha$ are referred to as different interpretations of SDEs with $\backslash alpha=1,1/2,0$ being respectively the Ito, Stratonovich, and Kolmogorov interpretations.

This ambiguity can be removed by additional conditions. In quantum theory, the condition is Hermiticity of Hamiltonian, which is satisfied by the Weyl symmetrization rule corresponding to $\backslash alpha=1/2$. In STS, the condition is that the SEO equals the GTO, which is also achieved at $\backslash alpha=1/2$. In other words, only the Stratonovich interpretation of SDEs is consistent with the dynamical systems theory approach. Other interpretations differ by the shifted flow vector field in the corresponding SEO, $F\_\backslash alpha\; =\; F\; -\; \backslash Theta(2\backslash alpha-1)(G\_a\backslash cdot\backslash partial)\; G\_a$.

File:STS Phase Diagram.png

The wavefunctions of STS depend not only on the original dynamical variables but also on their supersymmetric partners $\backslash chi$. These Grassmann numbers, or fermions, represent the differentials of the wavefunctions understood as differential forms. The fermions are intrinsically linked to stochastic Lyapunov exponents that define the butterfly effect. Therefore, it is believed that the effective field theory for these fermions -- referred to as goldstinos in the context of the spontaneous TS breaking -- is essentially a field theory of the butterfly effect.

The response of the model can be analyzed using the concept of generating functional:

$$G(\backslash eta)\; =\; -\backslash log\; \backslash lim\_\{T\backslash to\backslash infty\}\; \backslash langle\; g\; |\; \backslash hat\; M\; \_\{T/2,\; -T/2\}(\backslash eta)\; |\; g\; \backslash rangle,$$

where $\backslash eta$ denotes external probing fields, $\backslash hat\; M\; \_\{T/2,\; -T/2\}(\backslash eta)$ is the perturbed SEO/GTO, and $|\; g\; \backslash rangle$ 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, $\backslash text\{Re\; \}\; H\_g=\backslash min\backslash nolimits\_\{\backslash alpha\}\backslash text\{Re\; \}H\_\backslash alpha.$

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.

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.{{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}} The implications of complex $H\_g$ in a more general setting remain unexplored.

Since the late 80's,{{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}}{{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}}

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.

{{cite book|first=Markus|last=Aschwanden|title=Self-Organized Criticality in Astrophysics|year=2011|publisher =Springer}}

This phase has also been recognized as potentially significant for information processing.{{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}}{{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}}

Its phenomenological understanding is largely based on the concepts of self-adaptation and self-organization.{{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}}{{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 }}

STS offers the following explanation for the Edge of chaos (see figure on the right).{{Cite journal |last=Ovchinnikov |first=I.V. |title=Ubiquitous order known as chaos |date=2024-02-15 |journal=Chaos, Solitons & Fractals |language=en |volume=181 |issue=5 |pages=114611 |doi=10.1016/j.chaos.2024.114611 |bibcode=2024CSF...18114611O |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077924001620 |issn = 0960-0779}} 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.

{{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}}

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.

• Stochastic quantization
• Supersymmetric quantum mechanics
• Topological quantum field theory
• Stochastic differential equation
• Dynamical systems theory
• Chaos theory
• Self-Organized Criticality

{{Reflist}}

{{Industrial and applied mathematics}}

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