Monday, January 30, 2012

1201.5840 (Igor V. Ovchinnikov)

Dynamical Topological Symmetry Breaking as the Origin of Turbulence,
Non-Markovianity, and Self-Similarity

Igor V. Ovchinnikov
Here it is shown that the most general Parisi-Sourlas-Wu stochastic
quantization procedure applied to any stochastic differential equation (SDE)
leads to a Witten-type topological field theory - a model with a global
topological Becchi-Rouet-Stora-Tyutin supersymmetry (Q-symmetry). Q-symmetry
can be dynamically broken only by (anti-)instantons - ultimately nonlinear
sudden tunneling processes of (creation)annihilation of solitons, e.g.,
avalanches in self-organized criticality (SOC) or (creation)annihilation of
vortices in turbulent water. The phases with unbroken Q-symmetry are
essentially markovian and can be understood solely in terms of the conventional
Fokker-Plank evolution of the probability density. For these phases, Ito
interpretation of SDEs and/or Martin-Siggia-Rose approximation of the
stochastic quantization are applicable. SOC, turbulence, glasses, quenches etc.
constitute the "generalized turbulence" category of stochastic phases with
broken Q-symmetry. In this category, (anti-)instantons condense due to the
non-potential Novikov-type driving that can be interpreted as an external
energy reservoir. This dynamically mixes the probability distribution and the
wavefunctions of non-trivial Fadeev-Popov ghost content. Even for white noises,
the stochasticity can not be described in terms of the probability distribution
only. Such systems may be said to possess "genuine" non-Markovianity.
Representing instanton modulii and having the meaning of Ruelle-Pollicott
resonances, gapless goldstinos explain stochastic self-similarity of
generalized turbulence, e.g., algebraic statistics of avalanches in SOCs and
algebraic power spectrum of turbulent water. It is pointed out that stochastic
quantization is closely related to the concept of negative probabilities that
represent the freedom of the stochastic system to chose among various possible
solutions of SDE.
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