Abstract
Symmetries play paramount roles in dynamics of physical systems. All theories of quantum physics and microworld including the fundamental Standard Model are constructed on the basis of symmetry principles. In classical physics, the importance and weight of these principles are the same as in quantum physics: dynamics of complex nonlinear statistical systems is straightforwardly dictated by their symmetry or its breaking, as we demonstrate on the example of developed (magneto)hydrodynamic turbulence and the related theoretical models. To simplify the problem, unbounded models are commonly used. However, turbulence is a mesoscopic phenomenon and the size of the system must be taken into account. It turns out that influence of outer length of turbulence is significant and can lead to intermittency. More precisely, we analyze the connection of phenomena such as behavior of statistical correlations of observable quantities, anomalous scaling, and generation of magnetic field by hydrodynamic fluctuations with symmetries such as Galilean symmetry, isotropy, spatial parity and their violation and finite size of the system.
Highlights
The success of physics is to a large extent dictated by its enormous predictive power describing many natural phenomena
The crucial problem in many phenomena encountered in physics is the proper identification of underlying symmetry and the mechanism that leads to its violation
In non-equilibrium statistical physics, which deals with systems containing many interacting degrees of freedom, symmetries are present at different levels of theoretical description
Summary
The success of physics is to a large extent dictated by its enormous predictive power describing many natural phenomena. In both statistical mechanics and hydrodynamic transport problems, the interest in the IR behavior of statistical models is determined by the property of the basic field-theoretic tool—perturbation theory—to reproduce the observed singular behavior of certain physical quantities only in the limit of an infinite (flat) space This infrared limit is usually rather sensitive to the large-scale structure of the model and care has to exercised when passing to the limit. Symmetry and similarity arguments have allowed infering important conclusions about the scaling behavior of velocity correlation functions in the case of very large Reynolds numbers (the famous Kolmogorov theory in the first place) [22,23].
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