Supersymmetric theory of stochastic dynamics: Difference between revisions

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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 [[Spontaneous symmetry breaking|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 othermodern wordstheoretcal nomenclature, STSchaos, revealsalong thatwith chaosother realizations of [[spontaneous symmetry breaking]], is aan [[spontaneous long-rangesymmetry orderbreaking#Generalisation and technical usage|ordered phase]] -- a perspective anticipated in early discussions of [[complexity]]: as pointed out in Ref.the context of STS:<ref>{{cite journal

<ref>{{cite journal

| last = Uthamacumaran

| first = Abicumaran

Line 33 ⟶ 32:

| url = https://www.sciencedirect.com/science/article/pii/S2666389921000404

| access-date = 2025-06-05

}}</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...''

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.