A critical examination on the symmetries and their importance for statistical turbulence theory

A detailed theoretical investigation is given which demonstrates that a recently proposed set of statistical symmetries is physically void. Although they are mathematically admitted as unique symmetry transformations by the underlying statistical Navier-Stokes equations up to the functional level of the Hopf equation, by closer inspection, however, they lead to physical inconsistencies and erroneous conclusions in the theory of turbulence. These new statistical symmetries are thus misleading in so far as they form within an unmodelled theory a set of analytical results which at the same time lacks physical consistency. Our investigation will expose this inconsistency on different levels of statistical description, where on each level we will gain new insights for their non-physical transformation behavior. With a view to generate invariant turbulent scaling laws, the consequences will be finally discussed when trying to analytically exploit such symmetries. In fact, a mismatch between theory and numerical experiment is conclusively quantified. We ultimately propose a general strategy on how to not only track unphysical statistical symmetries, but also on how to avoid generating such misleading invariance results from the outset. All the more so as this specific study on physically inconsistent symmetries in turbulent flows only serves as a representative example within the broader context of statistical invariance analysis. In this sense our investigation is applicable to all areas of statistical physics in which symmetries get determined in order to either characterize complex dynamical systems, or in order to extract physically useful and meaningful information from the underlying dynamical process itself.