Liouville spectral gap and bifurcation-driven Lagrangian–Eulerian decoupling with nondiffusive turbulence closures

Abstract In fully developed homogeneous and isotropic turbulence, the Lagrangian and Eulerian descriptions of motion, although formally equivalent, become statistically decoupled. In this work, by invoking Liouville theorem, we show that the joint probability density function (PDF) of the Eulerian and Lagrangian fields, evolving from arbitrary initial conditions, relaxes exponentially toward a factorized form given by the product of the corresponding marginal PDFs. This relaxation is governed by a genuine spectral gap of the Liouville operator, whose magnitude is primarily set by the bifurcation rate of the velocity-gradient dynamics, whereas the contribution of Lyapunov exponents is shown to be significantly smaller. As a consequence, Eulerian–Lagrangian correlations decay rapidly, and if the joint PDF is initially factorized, its factorized structure is preserved at all subsequent times, with each marginal evolving independently under the corresponding dynamics. We further show that the formal equivalence between the two descriptions implies the invariance of the relative kinetic energy between arbitrarily chosen points. When combined with the asymmetric statistics of instantaneous finite-scale Lyapunov exponents in incompressible turbulence, this property provides a quantitative interpretation of particle-pair separation and of the turbulent energy cascade. Finally, these results naturally lead to nondiffusive closure relations for the von Kármán–Howarth and Corrsin equations, which coincide with those previously proposed by the author, thereby providing an independent theoretical validation of those closures.