Lagrangian-Eulerian Decoupling in Nondiffusive Turbulence: Liouville Spectral Gap and Bifurcation Dynamics

Although formally equivalent, Lagrangian and Eulerian descriptions statistically decouple in homogeneous isotropic turbulence. Applying Liouville's theorem, we prove that the joint PDF relaxes exponentially to a factorized state, driven by a spectral gap of the Liouville operator. This gap is primarily governed by the bifurcation rate of the velocity gradient, with a minor contribution from Lyapunov exponents. This decoupling ensures independent marginal dynamics and rapid correlation decay. Furthermore, the invariance of relative kinetic energy derived from this equivalence quantitatively explains particle separation and the turbulent cascade. These results yield nondiffusive closures for the von Kármán–Howarth and Corrsin equations, offering a new independent validation of existing theories.