Topological Supersymmetry Breaking as the Origin of the Butterfly Effect

Abstract

Previously, there existed no clear explanation why chaotic dynamics is always accompanied by the infinitely long memory of perturbations (and/or initial conditions) known as the butterfly effect (BE). In this paper, it is shown that within the recently proposed approximation-free supersymmetric theory of stochastic (partial) differential equations (SDE), the BE is a derivable consequence of (stochastic) chaos, a rigorous definition of which is the spontaneous breakdown of topological supersymmetry that all SDEs possess. It is also discussed that the concept of ergodicy must be refined under the condition of the spontaneous breakdown of pseudo-time-reversal symmetry when the model has "physical" states of multiple eigenvalues that survive the physical limit of the infinitely long temporal propagation.