2012/03/31 by Igor V. Ovchinnikov
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic number #Chaotic #Dynamical system (definition) #Dynamical systems theory #Fractal #Invariant (physics) #Quantum #Quantum field theory #Quantum many-body systems #Statistical Mechanics and Entropy #Symmetry breaking #hep-th #math-ph #math.MP #nlin.PS
paper · pdf · doi:10.1063/1.4746037
published as Chaos 22, 033134 (2012) · 18 pages, 4 figures, published version
openalex publication_date 2012/08/28 · arxiv created 2012/08/30 · arxiv updated 2012/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively stable due to what is generically known as non-renormalization theorems. As a result, all (equilibrium) dynamical models are divided into three major categories: Markovian models with unbroken Q-symmetry, chaotic models with Q-symmetry spontaneously broken on the mean-field level by, e.g., fractal invariant sets (e.g., strange attractors), and intermittent or self-organized critical (SOC) models with Q-symmetry dynamically broken by the condensation of instanton-antiinstanton configurations (earthquakes, avalanches, etc.) SOC is a full-dimensional phase separating chaos and Markovian dynamics. In the deterministic limit, however, antiinstantons disappear and SOC collapses into the "edge of chaos." Goldstone theorem stands behind spatio-temporal self-similarity of Q-broken phases known under such names as algebraic statistics of avalanches, 1/f noise, sensitivity to initial conditions, etc. Other fundamental differences of Q-broken phases is that they can be effectively viewed as quantum dynamics and that they must also have time-reversal symmetry spontaneously broken. Q-symmetry breaking in non-equilibrium situations (quenches, Barkhausen effect, etc.) is also briefly discussed.