Supersymmetric theory of stochastic dynamics: Difference between revisions

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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 [[Supersymmetry#Supersymmetry in dynamical systems|topological supersymmetry]] (TS) enabling the generalization of certain concepts from [[Deterministic system|deterministic]] to [[Stochastic process|stochastic]] models. At the same time, it can be looked upon as a [[Topological quantum field theories|topological field theory]] of stochastic dynamics that reveals various topological aspects.

STS seeks to give a rigorous mathematical derivation to several [[Universality class|universal]] phenomena of [[Stochastic process|stochastic dynamical systems]]. It identifies [[Spontaneous symmetry breaking|spontaneous breakdown of TS]], present in all stochastic models, as the stochastic generalization of [[chaos theory|chaos]]. In this view, STS proposes a notionreveals that dynamical chaos is a form of long-range [[Topological order|topological order]]. --The atheory perspectivealso longprovides anticipatedthe bylowest thelevel pioneersclassification of thestochastic conceptchaos ofwhich [[complexity]].has Asa pointedpotential outto inexplain Ref[[Self-organized criticality|self-organized criticality]].

== 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]]/[[State-space representation|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. In other words, STS reveals that chaos is a spontaneous long-range order -- a perspective long anticipated within the concept of complexity: as pointed out in Ref.

<ref>{{cite journal

| last = Uthamacumaran

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}}</ref> 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...''

STS provides a solid mathematical foundation for this picture and provides a potential explanation for [[Self-organized criticality|self-organized criticality]].

== 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]]/[[State-space representation|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 [[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 ==