A Two-Level Relative-Entropy Theory for Isotropic Turbulence Spectra: Fokker–Planck Semigroup Irreversibility and WKB Selection of Dissipation Tails

We propose a two-level theory that connects a Lin-equation-based dynamical coarse-graining of the turbulence cascade with an information-theoretic selection principle in logarithmic wavenumber space, thereby placing the dissipation-range spectral shape on a verifiable logical chain rather than an ad hoc fit. In the first (dynamical) stage, an autonomous conservative Fokker–Planck description is formulated for the normalized density and probability current; assuming sufficient boundary decay and a strictly positive effective diffusion, we prove that the sign-reversed KL divergence is a Lyapunov functional, yielding a rigorous H-theorem and fixing the arrow of time in scale space. In the second (selection) stage, the dissipation range is posed as a stationary boundary-value problem for an open system by introducing a killing term for an unnormalized scale density. WKB (Liouville–Green) analysis constrains the admissible tail class to a stretched-exponential form and links the tail exponent to the high-wavenumber scaling of the effective diffusion. To eliminate arbitrariness, the exponential prefactor is fixed by dissipation-rate consistency, and the remaining degree of freedom is identified via one-dimensional KL minimization (Hyper-MaxEnt) against a globally constructed reference distribution. The resulting exponent range is validated against high-resolution DNS spectra reported in the literature.