34 research outputs found
Symmetry Breaking in Stochastic Dynamics and Turbulence
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
Renormalization Group in Non-Relativistic Quantum Statistics
Dynamic behaviour of a boson gas near the condensation transition in the
symmetric phase is analyzed with the use of an effective large-scale model derived from
time-dependent Green functions at finite temperature. A renormalization-group analysis
shows that the scaling exponents of critical dynamics of the effective multi-charge model
coincide with those of the standard model A. The departure of this result from the descrip tion of the superfluid transition by either model E or F of the standard phenomenological
stochastic models is corroborated by the analysis of a generalization of model F, which
takes into account the effect of compressible fluid velocity. It is also shown that, con trary to the single-charge model A, there are several correction exponents in the effective
model, which are calculated at the leading order of the = 4 − d expansion
Field theoretic renormalization group for a nonlinear diffusion equation
The paper is an attempt to relate two vast areas of the applicability of the
renormalization group (RG): field theoretic models and partial differential
equations. It is shown that the Green function of a nonlinear diffusion
equation can be viewed as a correlation function in a field-theoretic model
with an ultralocal term, concentrated at a spacetime point. This field theory
is shown to be multiplicatively renormalizable, so that the RG equations can be
derived in a standard fashion, and the RG functions (the function and
anomalous dimensions) can be calculated within a controlled approximation. A
direct calculation carried out in the two-loop approximation for the
nonlinearity of the form , where is not necessarily
integer, confirms the validity and self-consistency of the approach. The
explicit self-similar solution is obtained for the infrared asymptotic region,
with exactly known exponents; its range of validity and relationship to
previous treatments are briefly discussed.Comment: 8 pages, 2 figures, RevTe