Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic dynamics on the intersection of dynamical systems theory, topological field theories, stochastic differential equations (SDE), and the theory of pseudo-Hermitian operators. It can be seen as an algebraic dual to the traditional set-theoretic framework of the dynamical systems theory, with its added algebraic structure and an inherent topological supersymmetry (TS) enabling the generalization of certain concepts from deterministic to stochastic models. Using tools of topological field theory originally developed in high-energy physics, STS seeks to give a rigorous mathematical derivation to several universal phenomena of stochastic dynamical systems. Particularly, the theory identifies dynamical chaos as a spontaneous order originating from the TS hidden in all stochastic models. STS also provides the lowest level classification of stochastic chaos which has a potential to explain self-organized criticality.
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/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 is chaotic because it exhibits a stochastic variant of the butterfly effect. In modern theoretical nomenclature, chaos, along with other realizations of spontaneous symmetry breaking, is an ordered phase—a perspective anticipated in early discussions of complexity: as pointed out in the context of STS:
... chaos is counter-intuitively the "ordered" phase of dynamical systems. Moreover, a pioneer of complexity, Prigogine, would define chaos as a spatiotemporally complex form of order... 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 has been introduced in the dynamical systems theory. This concept underlies the stochastic evolution operator of STS and provides it with a solid and natural 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
Generalized transfer operator 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:
x ˙ ( t ) = F ( x ( t ) ) + ( 2 Θ ) 1 / 2 G a ( x ( t ) ) ξ a ( t ) ≡ F ( ξ ( t ) ) , {\displaystyle {\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 a closed smooth manifold, X {\textstyle X} , called in dynamical systems theory a state space while in physics, where X {\displaystyle X} is often a symplectic manifold with half of variables having the meaning of momenta, it is called the phase space. Further, F ∈ T X {\displaystyle F\in TX} is a sufficiently smooth flow vector field from the tangent space of X {\displaystyle X} having the meaning of deterministic law of evolution, and G a ∈ T X , a = 1 , … , D ξ {\displaystyle G_{a}\in TX,a=1,\ldots ,D_{\xi }} is a set of sufficiently smooth vector fields that specify how the system is coupled to the time-dependent noise, ξ ( t ) ∈ R D ξ {\displaystyle \xi (t)\in \mathbb {R} ^{D_{\xi }}} , which is called additive/multiplicative depending on whether G a {\displaystyle G_{a}} 's are independent/dependent on the position on X {\displaystyle X} . The randomness of the noise will be introduced later. For now, the noise is a deterministic function of time and the equation above is an ordinary differential equation (ODE) with a time-dependent flow vector field, F {\displaystyle {\mathcal {F}}} . The solutions/trajectories of this ODE are differentiable with respect to initial conditions even for non-differentiable ξ ( t ) {\displaystyle \xi (t)} 's. In other words, there exists a two-parameter family of noise-configuration-dependent diffeomorphisms:
M ( ξ ) t t ′ : X → X , M ( ξ ) t t ′ ∘ M ( ξ ) t ′ t ″ = M ( ξ ) t t ″ , M ( ξ ) t t ′ | t = t ′ = Id X , {\displaystyle M(\xi )_{tt'}:X\to X,M(\xi )_{tt'}\circ M(\xi )_{t't''}=M(\xi )_{tt''},\left.M(\xi )_{tt'}\right|_{t=t'}={\text{Id}}_{X},} such that the solution of the ODE with initial condition x ( t ′ ) = x ′ {\displaystyle x(t')=x'} can be expressed as x ( t ) = M ( ξ ) t t ′ ( x ′ ) {\displaystyle x(t)=M(\xi )_{tt'}(x')} . The dynamics can now be defined as follows: if at time t ′ {\displaystyle t'} , the system is described by the probability distribution P ( x ) {\displaystyle P(x)} , then the average value of some function f : X → R {\displaystyle f:X\to \mathbb {R} } at a later time t {\displaystyle t} is given by:
f ¯ ( t ) = ∫ X f ( M ( ξ ) t t ′ ( x ) ) P ( x ) d x 1 ∧ . . . ∧ d x D = ∫ X f ( x ) M ^ ( ξ ) t ′ t ∗ ( P ( x ) d x 1 ∧ . . . ∧ d x D ) . {\displaystyle {\bar {f}}(t)=\int _{X}f\left(M(\xi )_{tt'}(x)\right)P(x)dx^{1}\wedge ...\wedge dx^{D}=\int _{X}f(x){\hat {M}}(\xi )_{t't}^{*}\left(P(x)dx^{1}\wedge ...\wedge dx^{D}\right).}
Here M ^ ( ξ ) t ′ t ∗ {\displaystyle {\hat {M}}(\xi )_{t't}^{*}} is action or pullback induced by the inverse map, M ( ξ ) t t ′ − 1 = M ( ξ ) t ′ t {\displaystyle M(\xi )_{tt'}^{-1}=M(\xi )_{t't}} , on the probability distribution understood in a coordinate-free setting as a top-degree differential form. Pullbacks are a wider concept, defined also for k-forms, i.e., differential forms of other possible degrees k, 0 ≤ k ≤ D = d i m X {\displaystyle 0\leq k\leq D=dimX} , ψ ( x ) = ψ i 1 . . . . i k ( x ) d x 1 ∧ . . . ∧ d x k ∈ Ω ( k ) ( x ) {\displaystyle \psi (x)=\psi _{i_{1}....i_{k}}(x)dx^{1}\wedge ...\wedge dx^{k}\in \Omega ^{(k)}(x)} , where Ω ( k ) ( x ) {\displaystyle \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,
| ψ ( t ) ⟩ = M ^ ( ξ ) t ′ t ∗ | ψ ( t ′ ) ⟩ , {\displaystyle |\psi (t)\rangle ={\hat {M}}(\xi )_{t't}^{*}|\psi (t')\rangle ,}
where | ψ ⟩ ∈ Ω ( X ) = ⨁ k = 0 D Ω ( k ) ( X ) {\displaystyle |\psi \rangle \in \Omega (X)=\bigoplus \nolimits _{k=0}^{D}\Omega ^{(k)}(X)} is a time-dependent "wavefunction", adopting the terminology of quantum theory. Unlike, say, trajectories or positions in X {\displaystyle X} , pullbacks are linear objects even for nonlinear X {\displaystyle 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. For Gaussian white noise, ⟨ ξ a ( t ) ⟩ noise = 0 , ⟨ ξ a ( t ) ξ b ( t ′ ) ⟩ noise = δ a b δ ( t − t ′ ) {\displaystyle \langle \xi ^{a}(t)\rangle _{\text{noise}}=0,\langle \xi ^{a}(t)\xi ^{b}(t')\rangle _{\text{noise}}=\delta ^{ab}\delta (t-t')} ..., the GTO is
M ^ t t ′ = ⟨ M ^ ( ξ ) t ′ t ∗ ⟩ noise = e − ( t − t ′ ) H ^ . {\displaystyle {\hat {\mathcal {M}}}_{tt'}=\langle {\hat {M}}(\xi )_{t't}^{*}\rangle _{\text{noise}}=e^{-(t-t'){\hat {H}}}.}
Here, the infinitesimal GTO is the stochastic evolution operator in the Stratonovich interpretation in the traditional approach to SDEs,
H ^ = L ^ F − Θ L ^ G a L ^ G a , {\displaystyle {\hat {H}}={\hat {L}}_{F}-\Theta {\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}},} where L ^ F {\displaystyle {\hat {L}}_{F}} is the Lie derivative along the vector field specified in the subscript. Note, however, that its fundamental mathematical meaning—the pullback averaged over noise—ensures that GTO is unique (see Operator representation below).
Topological supersymmetry With the help of Cartan formula, saying that Lie derivative is "d-exact", i.e., can be given as, e.g., L ^ A = [ d ^ , ı ^ A ] {\displaystyle {\hat {L}}_{A}=[{\hat {d}},{\hat {\imath }}_{A}]} , where square brackets denote bi-graded commutator and d ^ {\displaystyle {\hat {d}}} and ı ^ A {\displaystyle {\hat {\imath }}_{A}} are, respectively, the exterior derivative and interior multiplication along A, the following explicitly
can be obtained, where d ¯ ^ = ı ^ F − Θ ı ^ G a L ^ G a {\displaystyle {\hat {\bar {d}}}={\hat {\imath }}_{\mathcal {F}}-\Theta {\hat {\imath }}_{G_{a}}{\hat {L}}_{G_{a}}} . This form of the evolution operator is similar to that of Supersymmetric quantum mechanics, and it is a central feature of topological field theories of Witten-type. It assumes that the GTO commutes with d ^ {\displaystyle {\hat {d}}} , which is a (super)symmetry of the model. This symmetry is referred to as topological supersymmetry (TS), particularly because the exterior derivative plays a fundamental role in algebraic topology. TS pairs up eigenstates of GTO into doublets.
Eigensystem of GTO 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. This form of spectrum implies the presence of pseudo-time-reversal symmetry. Each eigenstate has a well-defined degree.
H ^ ( 0 , D ) {\displaystyle {\hat {H}}^{(0,D)}} do not break TS, min Re ( spec H ^ ( 0 , D ) ) = 0 {\displaystyle {\text{min Re}}(\operatorname {spec} {\hat {H}}^{(0,D)})=0} . Each De Rham cohomology provides one zero-eigenvalue supersymmetric "singlet" such that d ^ | θ ⟩ = 0 , ⟨ θ | d ^ = 0 {\displaystyle {\hat {d}}|\theta \rangle =0,\langle \theta |{\hat {d}}=0} . The singlet from H ^ ( D ) {\displaystyle {\hat {H}}^{(D)}} is the stationary probability distribution known as "ergodic zero". All the other eigenstates are non-supersymmetric "doublets" related by TS: H ^ | α ⟩ = H α | α ⟩ , H ^ | α ′ ⟩ = H α | α ′ ⟩ {\displaystyle {\hat {H}}|\alpha \rangle =H_{\alpha }|\alpha \rangle ,\;{\hat {H}}|\alpha '\rangle =H_{\alpha }|\alpha '\rangle } and ⟨ α | H ^ = ⟨ α | H α , ⟨ α ′ | H ^ = ⟨ α ′ | H α {\displaystyle \langle \alpha |{\hat {H}}=\langle \alpha |H_{\alpha },\langle \alpha '|{\hat {H}}=\langle \alpha '|H_{\alpha }} , where H α {\displaystyle H_{\alpha }} is the corresponding eigenvalue, and | α ′ ⟩ = d ^ | α ⟩ , ⟨ α | = ⟨ α ′ | d ^ {\displaystyle |\alpha '\rangle ={\hat {d}}|\alpha \rangle ,\;\langle \alpha |=\langle \alpha '|{\hat {d}}} .
Stochastic 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 t t ′ = T r M ^ t t ′ = ∑ α e − ( t − t ′ ) H α , {\displaystyle Z_{tt'}=Tr{\hat {\mathcal {M}}}_{tt'}=\sum \nolimits _{\alpha }e^{-(t-t')H_{\alpha }},}
grows exponentially in the limit of infinitely long evolution signaling the exponential growth of the number of closed solutions—the hallmark of chaotic dynamics. In terms of the infinitesimal GTO, this condition reads,
Δ = − min α Re H α > 0 , {\displaystyle \Delta =-\min _{\alpha }{\text{Re }}H_{\alpha }>0,}
where Δ {\displaystyle \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.
Stochastic Poincaré–Bendixson theorem Due to one of the spectral properties of GTO that H ^ ( 0 , D ) {\displaystyle {\hat {H}}^{(0,D)}} never break TS, i.e., min Re ( spec H ^ ( 0 , D ) ) = 0 {\displaystyle {\text{min Re}}(\operatorname {spec} {\hat {H}}^{(0,D)})=0} , a model has got to have at least two degrees other than 0 and D in order to accommodate a non-supersymmetric doublet with a negative real part of its eigenvalue and, consequently, be chaotic. This implies D = dim X ≥ 3 {\displaystyle D={\text{dim }}X\geq 3} , which can be viewed as a stochastic generalization of the Poincaré–Bendixson theorem.
Sharp trace and Witten Index Another object of interest is the sharp trace of the GTO,
W = T r ( − 1 ) k ^ M ^ t t ′ = ∑ α ( − 1 ) k α e − ( t − t ′ ) H α , {\displaystyle W=Tr(-1)^{\hat {k}}{\hat {\mathcal {M}}}_{tt'}=\sum \nolimits _{\alpha }(-1)^{k_{\alpha }}e^{-(t-t')H_{\alpha }},}
where k ^ | ψ α ⟩ = k α | ψ α ⟩ {\displaystyle {\hat {k}}|\psi _{\alpha }\rangle =k_{\alpha }|\psi _{\alpha }\rangle } with k ^ {\displaystyle {\hat {k}}} being the operator of the degree of the differential form. This is a fundamental object of topological nature known in physics as the Witten index. From the properties of the eigensystem of GTO, only supersymmetric singlets contribute to the Witten index, W = ∑ k = 0 D ( − 1 ) k B k = E u . C h ( X ) {\displaystyle W=\sum \nolimits _{k=0}^{D}(-1)^{k}B_{k}=Eu.Ch(X)} , where E u . C h . {\displaystyle Eu.Ch.} is the Euler characteristic and B 's are the numbers of supersymmetric singlets of the corresponding degree. These numbers equal Betti numbers as follows from one of the properties of GTO that each de Rham cohomology class provides one supersymmetric singlet.
Physical Perspective
Parisi–Sourlas method as a BRST gauge-fixing procedure 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. The path integral representation of the Witten index can be achieved in three steps: (i) introduction of the dynamical variables into the partition function of the noise; (ii) BRST gauge fixing the integration over the paths to the trajectories of the SDE which can be looked upon as the Gribov copies; and (iii) out integration of the noise. This can be expressed as the following
Here, the noise is assumed Gaussian white, p.b.c. signifies periodic boundary conditions, J ( ξ ) {\displaystyle \textstyle J(\xi )} is the Jacobian compensating (up to a sign) the Jacobian from the δ {\displaystyle \delta } -functional, Φ {\displaystyle \Phi } is the collection of fields that includes, besides the original field x {\displaystyle x} , the Faddeev–Popov ghosts χ , χ ¯ {\displaystyle \chi ,{\bar {\chi }}} and the Lagrange multiplier, B {\displaystyle B} , the topological and/or BRST supersymmetry is,
Q = ∫ d τ ( χ i ( τ ) δ / δ x i ( τ ) + B i ( τ ) δ / δ χ ¯ i ( τ ) ) , {\displaystyle Q=\textstyle \int d\tau (\chi ^{i}(\tau )\delta /\delta x^{i}(\tau )+B_{i}(\tau )\delta /\delta {\bar {\chi }}_{i}(\tau )),} that can be looked upon as a path integral version of exterior derivative, and the gauge fermion
Ψ = ∫ d τ ( ı x ˙ − d ¯ ) {\textstyle \Psi =\int d\tau (\imath _{\dot {x}}-{\bar {d}})} with
d ¯ = ı F − Θ ı G a L G a , and L G a = ( Q , ı G a ) {\textstyle \textstyle {\bar {d}}=\textstyle \imath _{F}-\Theta \imath _{G_{a}}L_{G_{a}},{\text{ and }}L_{G_{a}}=(Q,\imath _{G_{a}})} and ı A = i χ ¯ A {\textstyle \imath _{A}=i{\bar {\chi }}A} being the path integral versions of the Lie derivative and interior multiplication.
STS as a topological field theory The Parisi-Sourlas method is peculiar in that sense that it looks li