What question did this study set out to answer?

To develop a robust theory connecting turbulence spectra dynamics with information theory, focusing on dissipation-range characteristics.

February 12, 2026Open Access

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

Key Points

To develop a robust theory connecting turbulence spectra dynamics with information theory, focusing on dissipation-range characteristics.
Formulated a conservative Fokker-Planck equation for normalized density and probability currents.
Established a Lyapunov functional using Kullback-Leibler divergence to determine time direction in scale space.
Introduced a killing term for a stationary boundary-value problem to address dissipation range of an open system.
Conducted a WKB analysis to link tail exponents to effective diffusion scaling advocacy for stretched-exponential forms.
Validated the derived tail exponent range against high-resolution direct numerical simulation (DNS) spectra from literature.
Demonstrated the applicability of the H-theorem to establish irreversible dynamics in turbulence spectra.

Abstract

We propose a two-level theory that connects Lin-equation-based dynamical coarse-graining of the turbulence cascade with an information-theoretic selection principle in logarithmic wavenumber space. This framework places the dissipation-range spectral shape on a verifiable logical basis rather than on ad hoc fitting. At the first (dynamical) level, we formulate an autonomous conservative Fokker–Planck equation for the normalized density and probability current. Under sufficient boundary decay and a strictly positive effective diffusion, the sign-reversed Kullback–Leibler divergence is shown to be a Lyapunov functional, yielding a rigorous H-theorem and fixing the arrow of time in scale space. At the second (selection) level, the dissipation range is treated as a stationary boundary-value problem for an open system by introducing a killing term for an unnormalized scale density. A WKB (Liouville–Green) analysis restricts the admissible tail to a stretched-exponential form and links the tail exponent to the high-wavenumber scaling of the effective diffusion. The exponential prefactor is fixed by dissipation-rate consistency, and the remaining degree of freedom is determined by one-dimensional Kullback–Leibler minimization (Hyper-MaxEnt) against a globally constructed reference distribution. The resulting exponent range is validated against the high-resolution DNS spectra reported in the literature.

Bookmark

View Full Paper