Theory of multi-point probability densities for incompressible Navier-Stokes fluids

Abstract

An open problem arising in the statistical description of turbulence is related to the \textit{theoretical prediction based on first principles} of the so-called multi-point velocity probability density functions (PDFs) characterizing a Navier-Stokes fluid. In this paper it will be shown that - based on a suitable axiomatic approach - a solution to this problem can actually be achieved based on the so-called inverse kinetic theory (IKT), recently developed for incompressible fluids. More precisely, we intend to show, based on the requirement that \textit{the Boltzmann-Shannon entropy for the s-point velocity PDF ($f_s$) is independent of the order $s$ and is also maximal at all times}, that all multi-point PDFs are \textit{necessarily factorized in terms of the corresponding 1-point velocity PDF} ($f_1$). As a consequence the multi-point PDFs usually considered for the phenomenological description of turbulence can be theoretically predicted \textit{based on the knowledge of $% f_1$ achieved by means of IKT.