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 space|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 is chaotic because it exhibits a stochastic variant of the butterfly effect. In modern theoretcaltheoretical nomenclature, chaos, along with other realizations of [[spontaneous symmetry breaking]], is an [[spontaneous symmetry breaking#Generalisation and technical usage|ordered phase]] -- a perspective anticipated in early discussions of [[complexity]]: as pointed out in the context of STS:<ref>{{cite journal

| last = Uthamacumaran

| first = Abicumaran