Symmetry of the Lundgren–Monin–Novikov Equation for the Probability Distribution of the Vortex Field

Abstract

Polyakov [Nuclear Phys. B, 396, 1993] proposed a program for expanding the group of symmetries of hydrodynamic models to the conformal invariance of statistics in inverse cascades, where the conformal group is infinite-dimensional. This study presents group G of transformations of the equation for an \(n\)-point probability density function f_n (PDF) from an infinite chain of the Lundgren–Monin–Novikov equations (the statistical form of the Euler equations) for the field of a vortex of two-dimensional flow. The main result obtained is that the group G transforms conformally the characteristics of the zero-vorticity equation and, invariantly, the family of the f_n equations for the PDF along these lines. Along with other characteristics, the equation is not invariant. The action of G retains the PDF class. The results obtained can be used to study the invariance of the statistical properties in optical turbulence.