Supersymmetric theory of stochastic dynamics
By describing chaos as a spontaneous breakdown of topological supersymmetry, STS aims to explain long-range dynamical phenomena such as 1/f, flicker, and crackling noises and the power-law statistics of instantonic processes like earthquakes and neuroavalanches.
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",[1] whose evolution determines Lyapunov exponents.
From an algebraic topology perspective, the wavefunctions are differential forms[3] and dynamical systems theory defines their dynamics by the generalized transfer operator (GTO)[4][5] -- the pullback or action induced by the system on differential forms, averaged over noise.
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 first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by Giorgio Parisi and Nicolas Sourlas,[6][1] where Langevin SDEs -- SDEs with linear phase spaces, gradient flow vector fields, and additive noises -- were given supersymmetric representation with the help of the BRST gauge fixing procedure.
The original goal of their work was dimensional reduction, i.e., a specific cancellation of divergences in Feynman diagrams proposed a few years earlier by Amnon Aharony, Yoseph Imry, and Shang-keng Ma.
[7] Since then, the so-emerged supersymmetry of Langevin SDEs has been addressed from a few different angles [8][9][10][11][12] including the fluctuation dissipation theorems,[11] Jarzynski equality,[13] Onsager principle of microscopic reversibility,[14] solutions of Fokker–Planck equations,[15] self-organization,[16] etc.
The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including classical mechanics,[17][18] its stochastic generalization,[19] and higher-order Langevin SDEs.
[12] The theory of pseudo-Hermitian supersymmetric operators [20] and the relation between the Parisi-Sourlas method and Lyapunov exponents [2] 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.
[4][5] This concept underlies the stochastic evolution operator of STS and provides it with a solid mathematical meaning.
[22][23] The Parisi-Sourlas method has been recognized [24][17] as a member of Witten-type or cohomological topological field theory,[25][26][27][28][3][29][30][31] a class of models to which STS also belongs.
From the point of view of the theory of SDEs, this GTO is a stochastic evolution operator (SEO) in Stratonovich interpretation.
However, unlike SEOs in the theory of SDEs and/or the Parisi-Sourlas approach, the GTO has a clear-cut mathematical meaning, making it unique and eliminating the need for an additional interpretation beyond its definition.
In physical terms, this indicates the presence of a symmetry or, more precisely, a supersymmetry due to the nilpotency of the exterior derivative:
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.
In this formalism, the Q-exact pieces like the action of the Parisi-Sourlas approach serve as gauge fixing tools.
Different solutions at a fixed noise can be understood as Gribov copies and the fermions of the theory can be identified as Faddeev–Popov ghosts.
The 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
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 Parisi-Sourlas path integral with open boundary conditions is the stochastic evolution operator (SEO).
In quantum theory, the condition is the requirement for a Hermitian Hamiltonian, which is satisfied by the Weyl symmetrization rule corresponding to
At the same time, other interpretations are important in the context of discrete-time stochastic evolution and numerical implementation of SDEs.
[3] The fermions are intrinsically linked to stochastic Lyapunov exponents, [2] This suggests, particularly, that under conditions of spontaneous TS breaking, the effective theory for these fermions -- referred to as goldstinos in this context -- is essentially a theory of the butterfly effect.
The 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,
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.
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.
[42] Under this condition, the dynamics must be dominated by instantons with power-law distributions, as dictated by the Goldstone theorem.
