Topological field theory of dynamical systems

Details Topological field theory of dynamical systems Topological field theory of dynamical systems

2012-03-23

arxiv.org/abs/1203.5162v1

Physics

Mathematical Physics

8 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1201.5840

Scientific paper

Here, it is shown that the path-integral representation of any stochastic or deterministic dynamical system is a cohomological topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As a result, all dynamical systems are divided into two major categories: systems with broken and unbroken Q-symmetry. In deterministic systems, Q-symmetry can be spontaneously broken by the existence of a fractal invariant set. In this case the (deterministic) dynamics is chaotic. In stochastic systems, Q-symmetry can be dynamically broken prior to the chaotic regime by the condensation of instantons and anti-instantons. This situation corresponds to self-organized criticality, which is a full-dimensional phase that, however, collapses into "edge of chaos" in deterministic limit. Chaotic and self-organized critical systems exhibit self-similarity and non-Markovianity due to gapless goldstinos that must accompany spontaneous Q-symmetry breaking. In this sense, all the other systems, where Q-symmetry is unbroken, can be called Markovian.

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