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

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.

1. Introduction

The concept of the energy cascade in three-dimensional homogeneous isotropic turbulence originates from the seminal work of Kolmogorov [1]. In this picture, energy injected at large scales by external forcing is transferred through a hierarchy of interacting eddies toward smaller scales and ultimately dissipated by viscosity. Under the assumptions of self-similarity and local isotropy, Kolmogorov derived the inertial-range scaling $E\left(k\right)=C_K{\epsilon }^{2/3}{k}^{-5/3}$, which has become a cornerstone of turbulence theory. This framework was subsequently developed in Batchelor’s theory of homogeneous turbulence [2] and systematically reviewed in classical monographs and textbooks [3,4,5].

Despite its success, Kolmogorov’s theory is intrinsically confined to the inertial subrange and provides no prediction for the spectral shape outside it, particularly in the dissipation range. To address this limitation, spectral theories based on the Lin and Kármán–Howarth equations were developed, beginning with the pioneering analyses of Kovasznay [6] and Kraichnan [7], and later extended through closure approaches such as the direct-interaction approximation. While these theories offer deep insight into nonlinear energy transfer, they do not uniquely determine the functional form of the energy spectrum over the full range of wavenumbers.

As practical models that include the dissipation range, semi-empirical spectral forms such as Pao’s model [8] and Pope’s empirical spectrum [9] have been widely adopted. These models preserve the inertial-range $k^{-5/3}$ scaling while introducing an exponential cutoff near the Kolmogorov wavenumber $k_d$. They reproduce DNS results with remarkable accuracy [10,11]. However, the dissipation-range decay is postulated a priori and is not derived directly from the Navier–Stokes equations.

Recent high-Reynolds-number DNS and experimental studies have shown that the dissipation-range spectrum is not well represented by a simple exponential. Instead, it is more accurately described by a stretched-exponential form, $E\left(k\right)\sim C_K{\epsilon }^{2/3}{k}^{-5/3}\mathrm{e}\mathrm{x}\mathrm{p}[-\beta (k/k_d{)}^m]$, with an exponent typically in the range $m\in [4/3,\hspace{0.17em}3/2]$ [12,13,14]. The apparent dependence of $m$ on flow conditions and Reynolds number challenges the traditional notion of universality in turbulence theory. It raises several unresolved questions: Why does the dissipation-range spectrum take a stretched-exponential form? Why is the exponent confined to a narrow interval? What selection principle determines $m$? Can $m$ be fixed from Navier–Stokes dynamics without empirical fitting? Existing spectral models, which are largely phenomenological or semi-empirical, do not provide first-principles answers to these questions [15,16].

In parallel, turbulence has increasingly been examined from an information-theoretic perspective. Entropy-like measures have been introduced to quantify the formation and destruction of coherent structures [17]. Universal bounds on inter-scale information flow have been established in shell-model turbulence [18]. Concepts from quantum information theory, such as von Neumann and entanglement entropies, have also been applied to multi-field turbulence to characterize nonlinear mode coupling and energy-transfer directionality [19]. Together, these studies emphasize that energy dissipation and information generation are intrinsically linked. However, most information-theoretic approaches focus on measuring information flows or geometric structures, rather than on determining the functional form of the energy spectrum itself.

As a consequence, a fundamental gap remains in turbulence spectral theory. At present, there is no framework that simultaneously (i) establishes irreversibility through a coarse-grained dynamics derived from the Lin equation, (ii) rigorously constrains the admissible asymptotic form of the dissipation-range spectrum, and (iii) selects the realized spectral exponent through an information-theoretic variational principle, all within a logically closed and verifiable structure.

The present study addresses this gap by proposing a two-level entropy-based framework that explicitly separates dynamical admissibility from information-theoretic selection. At the first level (the dynamical level), the Lin equation is coarse-grained into an autonomous conservative Fokker–Planck description in logarithmic wavenumber space, $\xi =\mathrm{l}\mathrm{n}(k/k_0)$. By explicitly specifying boundary conditions and coefficient regularity, in particular, a strictly positive diffusion coefficient, the relative turbulence entropy is shown to act as a Lyapunov functional. This establishes a rigorous H-theorem and fixes an arrow of time in scale space.

At the second level (the selection level), the dissipation range is treated by formulating an effective stationary boundary-value problem that incorporates killing (absorption). A WKB (eikonal) analysis shows that the high-wavenumber tail must belong to a stretched-exponential class and yields a coupling relation between the tail exponent and the asymptotic scaling of the effective diffusion [20]. To enable a well-posed information-theoretic comparison, a physically meaningful lower bound ${\xi }_{-}=\mathrm{l}\mathrm{n}{k}_f$ is introduced, where $k_f$ denotes a representative forcing-band wavenumber. The domain is restricted to $\left.\xi \in [{\xi }_{-},\mathrm{\infty }\right)$, thereby excluding the energy-containing range from the present modeling. Incorporating the WKB-derived coupling as a constraint reduces the remaining freedom to a single parameter, which is uniquely determined through one-dimensional minimization of the Kullback–Leibler divergence (a reduced Hyper-MaxEnt principle).

Within this framework, the dissipation-range exponent $m^{\ast }$ is not introduced as an empirical fitting parameter. It emerges from a transparent logical sequence: irreversible structure at the dynamical level, asymptotic admissibility enforced by WKB analysis, reduction of degrees of freedom, and information-theoretic selection. The exponential dissipation spectrum is thus positioned as a consequence of first-principles reasoning rather than a phenomenological assumption.

2. Dynamical Modeling of the Turbulent Cascade

This section formulates the first stage (the dynamical level) of the proposed two-level theory. Throughout this work, we focus on statistically stationary, homogeneous, and isotropic turbulence at sufficiently high Reynolds numbers, where an inertial cascade in wavenumber space is well developed. These assumptions justify the use of the Lin equation and the subsequent coarse-grained description in logarithmic wavenumber space.

Accordingly, the present framework is not intended to describe strongly anisotropic flows, wall-bounded turbulence, or flows dominated by mean shear, rotation, or stratification, in which additional physical mechanisms alter the cascade dynamics. Instead, the theory is designed to capture the universal features of inter-scale energy transfer in an idealized isotropic setting, providing a controlled baseline from which extensions to more complex flow configurations may be developed.

We begin with the energy-spectrum balance equation derived from the Navier–Stokes equations, namely the Lin equation. The associated energy flux is intrinsically nonlocal, as it is defined through an integral over all wavenumbers arising from triadic interactions, and therefore admits no closed-form expression in general. In this study, the turbulent cascade is viewed as an aggregate of many local transfer events and is coarse-grained as an effective Markov process in logarithmic wavenumber space.

This viewpoint is conceptually related to early efforts to describe isotropic turbulence using generalized momentum-transfer and spectral transport theories, such as the approach proposed by Edwards and McComb [21]. In the continuous limit, the coarse-grained description leads to a Fokker–Planck (FP) equation, which provides a unified foundation for the probability current, entropy, and WKB analyses developed in subsequent sections. A central point is that the drift and diffusion coefficients $A(\xi ,t)$ and $D(\xi ,t)$ introduced here are effective quantities. They are defined as low-order moments of a coarse-grained transition kernel, rather than as exact quantities that fully encode triadic interactions. All subsequent analysis is constructed self-consistently within this effective-theory framework [6,7,21,22,23,24,25,26,27]

2.1. Lin Equation and the Energy-Spectrum Budget

Let $E(k,t)$ denote the energy spectrum of three-dimensional homogeneous isotropic turbulence, and define the total kinetic energy per unit mass by

$$E_{\mathrm{t}\mathrm{o}\mathrm{t}}(t)={\int }_0^{\mathrm{\infty }}E(k,t)\hspace{0.17em}dk.$$

Let $T_f(k,t)$ be the spectral density of energy injection by external forcing, and let $\nu$ be the kinematic viscosity. Then, $E(k,t)$ satisfies the Lin equation:

$${\displaystyle \frac{\partial E(k,t)}{\partial t}}+{\displaystyle \frac{\partial \mathsf{\Pi }(k,t)}{\partial k}}=T_f\left(k,t\right)-2\nu k^{2}E\left(k,t\right).$$

(1)

Here $\mathsf{\Pi }(k,t)$ is the energy flux in wavenumber space; since it arises from triadic interactions, it is, in general, defined by a nonlocal integral. In a statistically stationary state, ${\partial }_{t}E=0$, and (1) reduces to

$${\displaystyle \frac{\partial \mathsf{\Pi }\left(k\right)}{\partial k}}=T_f\left(k\right)-2\nu k^{2}E\left(k\right).$$

(2)

Define the dissipation rate $\epsilon$ by

$$\epsilon =2\nu {\int }_0^{\mathrm{\infty }}{k}^{2}E\left(k\right)\hspace{0.17em}dk.$$

(3)

If the forcing is localized in a finite wavenumber band, and in the inertial subrange $k_f\ll k\ll k_d$ one has $T_f\simeq 0$ while viscous dissipation is negligible, then (2) yields

$$\mathsf{\Pi }\left(k\right)\simeq \epsilon \left(k_f\ll k\ll k_d\right).$$

(4)

Consistently, Kolmogorov’s $-5/3$ law holds:

$$E\left(k\right)\simeq C_K\hspace{0.17em}{\epsilon }^{{\displaystyle \frac{2}{3}}}{k}^{-{\displaystyle \frac{5}{3}}}.$$

(5)

To connect with the dissipation range, define the Kolmogorov wavenumber by

$$k_d={\left({\displaystyle \frac{\epsilon }{{\nu }^3}}\right)}^{1/4}.$$

(6)

and introduce the dimensionless wavenumber $x=k/k_d$.

2.2. Unified Spectral Representation Including the Dissipation Range

To represent the full spectrum, including the dissipation range, in a unified form, we adopt the following cutoff representation:

$$E\left(k\right)=C_K\hspace{0.17em}{\epsilon }^{{\displaystyle \frac{2}{3}}}{k}^{-{\displaystyle \frac{5}{3}}}f\hspace{0.17em}\left({\displaystyle \frac{k}{k_d}}\right).$$

(7)

Here $f\left(x\right)$ is a smooth function satisfying $f\left(x\right)\to 1$ for $x\ll 1$ and decaying exponentially for $x\gg 1$. In the latter part of this paper (Section 4 and Section 5), we will constrain the dynamically admissible class of $f$ via asymptotic analysis and select the remaining freedom by an information-theoretic principle.

2.3. Log-Wavenumber Space and the Definition of Probability Densities

To treat the distribution in wavenumber space on a logarithmic scale, fix a reference wavenumber $k_0>0$ and introduce

$$\xi =\ln \hspace{0.17em}\left({\displaystyle \frac{k}{k_0}}\right).$$

(8)

The mapping $k=k_0{e}^{\xi }$ is a $C^{\mathrm{\infty }}$-diffeomorphism between $\mathbb{R}$ and $\left(0,\mathrm{\infty }\right)$, with $dk=k\hspace{0.17em}d\xi$.

Here, the assumption that the mapping $k$ ↦ $\xi =\ln \left(k/k_0\right)$ is a $C^{\mathrm{\infty }}$-diffeomorphism means that the change of variables between $k$-space and log-wavenumber space is smooth with a smooth inverse, and that derivatives of arbitrary order exist and are continuous. This regularity is required in the subsequent analysis to ensure that derivatives with respect to $\xi$ are well defined, that integration by parts and change-of-variable formulas used in the derivation of the Fokker–Planck equation are justified, and that higher-order asymptotic expansions, such as the WKB analysis in the dissipation range, are mathematically consistent. Henceforth, for each time $t$, we assume that the spectrum $E(k,t)$ is measurable on $\left(0,\mathrm{\infty }\right)$ and satisfies

$$E\left(k,t\right)\ge 0 (\mathrm{a}.\mathrm{e}. \mathrm{on} (0,\mathrm{\infty })).$$

(9)

Define the total kinetic energy per unit mass by

$$E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)={\int }_0^{\mathrm{\infty }}E\left(k,t\right)\hspace{0.17em}dk{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}k\hspace{0.17em}E\left(k,t\right)\hspace{0.17em}d\xi .$$

(10)

Define the unnormalized scale density in log-wavenumber space by

$$P\left(\xi ,t\right)\equiv k\hspace{0.17em}E\left(k,t\right),\left(k=k_0{e}^{\xi }\right)$$

(11)

By (9), $\overline{P}(\xi ,t)\ge 0$ (a.e.), and by (10) one has $\overline{P}(\cdot ,t)\in L^1\left(\mathbb{R}\right)$ with

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathcal{P}\left(\xi ,t\right)\hspace{0.17em}d\xi =E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)$$

(12)

We then define the normalized probability density in log-wavenumber space by

$$P\left(\xi ,t\right)\equiv {\displaystyle \frac{\mathcal{P}\left(\xi ,t\right)}{E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)}}={\displaystyle \frac{k\hspace{0.17em}E\left(k,t\right)}{E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)}},\left(k=k_0{e}^{\xi }\right).$$

(13)

By assumption, $0<E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)<\mathrm{\infty }$, the right-hand side is well-defined, and (12) immediately yields the normalization ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(\xi ,t)\hspace{0.17em}d\xi =1.$ Conversely, from (13), for any $k>0$,

$$E\left(k,t\right)={\displaystyle \frac{E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)}{k}}\hspace{0.17em}P\left(\xi ,t\right),\xi =\ln \hspace{0.17em}\left({\displaystyle \frac{k}{k_0}}\right).$$

(14)

From Section 2 through Section 3, $P(\xi ,t)$ denotes the normalized probability density defined in (13), while the unnormalized quantity is always denoted by $\overline{P}(\xi ,t)$. When, from Section 4 onward, the unnormalized density becomes the primary unknown, we continue to use the same symbols $P$ and $\overline{P}$ accordingly, without redefining $P$.

2.4. Coarse-Graining: From a Jump Process to the Fokker–Planck Equation

The energy flux $\mathsf{\Pi }(k,t)$ appearing in the Lin equation originates from triadic interactions and is, in general, defined as a nonlocal integral over the entire wavenumber range. In this section, rather than retaining the full nonlocal detail, we coarse-grain the cascade as an effective transition process on the logarithmic wavenumber $\xi$, and derive, in its diffusion limit, a Fokker–Planck (FP) equation. Using a Chapman–Kolmogorov-type master equation as the starting point is not in conflict with the section title “Fokker–Planck equation”: the master equation is the fundamental evolution law for a jump process, and in the continuous limit where near-neighbor jumps (small $\mathsf{\Delta }\xi$) dominate, its generator converges to a second-order differential operator, yielding the FP equation.

2.4.1. Assumptions for the Jump Process and the Master Equation

We approximate energy transfer on the logarithmic wavenumber axis as an accumulation of “jumps from $\xi$ to $\xi +\mathsf{\Delta }\xi$,” and represent the transition by a conditional transition kernel (jump kernel)

$$W(\mathsf{\Delta }\xi ;\xi ,t)\ge 0.$$

(15)

As explicit premises of the coarse-graining, we impose the following.

First, quasi-locality (locality): for fixed $\left(\xi ,t\right)$, the dominant transitions are concentrated in $\mid \mathsf{\Delta }\xi \mid \ll 1$, and large jumps are sufficiently suppressed. Introducing a small parameter $\delta \to 0$, this may be expressed, for instance, as

$${\int }_{\mid \mathsf{\Delta }\xi \mid >\eta }W(\mathsf{\Delta }\xi ;\xi ,t)\hspace{0.17em}d\left(\mathsf{\Delta }\xi \right)\underset{\delta \to 0}{\to }0 \left(\forall \hspace{0.17em}\eta >0\right),$$

(16)

which signifies the vanishing of the large-jump tail probability. This quasi-locality assumption in logarithmic wavenumber space is strongly supported by both theoretical and numerical studies of turbulent energy transfer, which demonstrate that the dominant contribution to the energy flux arises from interactions among comparable scales. Classical arguments due to Kraichnan and subsequent rigorous analyses have established the scale-locality of the inertial-range cascade, while direct numerical simulations have confirmed that nonlocal interactions provide only subleading contributions to the net flux [7,25,26,28].

Second, effective Markov property (short memory): on the coarse-grained timescale, short memory holds, and the time evolution of $\xi$ follows a Chapman–Kolmogorov-type master equation [23,29,30]. For the normalized probability density $P(\xi ,t)$ defined in Section 2.3, the master equation takes the form

$${\partial }_{t}P(\xi ,t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left[W\left(\mathsf{\Delta }\xi ;\xi -\mathsf{\Delta }\xi ,t\right)\hspace{0.17em}P\left(\xi -\mathsf{\Delta }\xi ,t\right)-W\left(\mathsf{\Delta }\xi ;\xi ,t\right)\hspace{0.17em}P\left(\xi ,t\right)\right]\hspace{0.25em} d\left(\mathsf{\Delta }\xi \right).$$

(17)

The right-hand side is the difference between the “probability inflow into $\xi$” and the “probability outflow from $\xi$,” and can be interpreted as the jump-process analog of the probability-conservation law.

2.4.2. Kramers–Moyal Expansion and the Diffusion Limit: Reduction to the FP Equation

To obtain a continuous-limit equation from (17), we perform a Kramers–Moyal (K–M) expansion using $\mid \mathsf{\Delta }\xi \mid \ll 1$. For this purpose, we assume that the conditional moments of the jump kernel,

$$a_n\left(\xi ,t\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\mathsf{\Delta }\xi {)}^n\hspace{0.17em}W\left(\mathsf{\Delta }\xi ;\xi ,t\right)\hspace{0.17em}d\left(\mathsf{\Delta }\xi \right),$$

(18)

exist and are finite up to the required order. Moreover, under tail suppression such as (16), higher-order remainder terms are controlled, and the interchange of expansion and integration (at least in a weak sense) is justified. Under this assumption, and in the diffusion approximation, the master equation reduces to a Fokker–Planck equation.

The Kramers–Moyal expansion and its diffusion approximation are standard tools in the theory of stochastic processes and Fokker–Planck equations [22,23]. Under these conditions, (17) reduces to the K–M expansion.

$${\partial }_{t}P={\sum }_{n=1}^{\mathrm{\infty }}{\displaystyle \frac{{\left(-1\right)}^n}{n!}}\hspace{0.17em}{\partial }_{\xi }^{\hspace{0.17em}n}\hspace{0.17em}\left(a_n\left(\xi ,t\right)\hspace{0.17em}P\left(\xi ,t\right)\right).$$

(19)

We then adopt, as a diffusion approximation, a second-order closure of (19). A crucial point is that, for a finite-order truncation to remain consistent with the nonnegativity of the probability density, the second-order closure has a distinguished status: in light of Pawula’s consistency result, the minimal nontrivial closure compatible with positivity is of second order. Accordingly, we employ the effective model assumption.

$$a_n\left(\xi ,t\right)\equiv 0 \left(n\ge 3\right).$$

(20)

Then only terms up to second order remain in (19), yielding the Fokker–Planck equation:

$${\partial }_{t}P\left(\xi ,t\right)=-{\partial }_{\xi }\hspace{0.17em}\left(A\left(\xi ,t\right)\hspace{0.17em}P\left(\xi ,t\right)\right)+{\partial }_{\xi }^{\hspace{0.17em}2}\hspace{0.17em}\left(D\left(\xi ,t\right)\hspace{0.17em}P\left(\xi ,t\right)\right).$$

(21)

The coefficients are defined by

$$A\left(\xi ,t\right):=a_1\left(\xi ,t\right),D\left(\xi ,t\right):={\displaystyle \frac{1}{2}}\hspace{0.17em}{a}_2\left(\xi ,t\right),$$

(22)

where $A$ is the effective drift and $D$ the effective diffusion coefficient. Thus, $A$ and $D$ do not represent the full details of triadic interactions; rather, they encode the “retained information” as low-order moments of the transition kernel $W$.

The physical meaning of these coefficients can be summarized as follows. The drift $A(\xi ,t)$ represents the mean directional transport of energy along the logarithmic wavenumber axis and thus reflects the net direction and intensity of the turbulent cascade. In contrast, the diffusion coefficient $D(\xi ,t)$ measures the statistical spread of scale-to-scale energy transfer induced by triadic interactions and encodes the degree of fluctuation and effective nonlocality retained under the coarse-graining.

In the inertial subrange, where the energy flux is approximately constant and scale invariance holds, $A(\xi ,t)$ is expected to be slowly varying and may be approximated as nearly constant, while $D(\xi ,t)$ depends, at most, weakly on $\xi$. In the dissipation range, viscous effects progressively suppress the cascade, leading to a decay of both $A(\xi ,t)$ and $D(\xi ,t)$. Near the crossover between the inertial and dissipation ranges, we do not impose any specific functional form for these coefficients; instead, we assume only sufficient smoothness to permit asymptotic analysis. This minimal assumption is essential for the subsequent WKB treatment, which constrains the admissible dissipation tails without prescribing their detailed scaling behavior a priori.

To write (21) in conservative form, introduce the probability current (probability flux) $J(\xi ,t)$ by

$${\partial }_{t}P\left(\xi ,t\right)=-{\partial }_{\xi }J\left(\xi ,t\right),$$

(23)

$$J\left(\xi ,t\right)=A\left(\xi ,t\right)\hspace{0.17em}P\left(\xi ,t\right)-{\partial }_{\xi }\hspace{0.17em}\left(D\left(\xi ,t\right)\hspace{0.17em}P\left(\xi ,t\right)\right).$$

(24)

In (24), the current is given as the drift contribution minus the diffusive-gradient contribution; we fix this sign convention throughout the paper.

To treat rigorously the $H$-theorem for conservative FP semigroups (monotonicity of relative entropy) in Section 3, (23) and (24) must be formulated within a framework that preserves mass (normalization). We therefore assume, first, that the diffusion coefficient admits a strictly positive lower bound on the domain of interest:

$$D(\xi ,t)\ge d_0>0.$$

(25)

This condition is essential to render later nonnegative expressions of the form $J^2/\left(DP\right)$ mathematically well-defined. From a physical standpoint, the assumptions of autonomous coefficients and strictly positive diffusion should be understood as properties of an effective, coarse-grained description of turbulence rather than of the instantaneous Navier–Stokes dynamics. While real turbulent flows may exhibit temporal variability and intermittency, the effective diffusion coefficient associated with scale-to-scale energy transfer is expected to remain positive when averaged over the cascade timescale relevant to the Fokker–Planck description. The autonomy and positivity assumptions are therefore not meant to represent exact microscopic constraints, but rather to ensure mathematical well-posedness of the conservative semigroup and to provide a controlled setting in which irreversibility can be rigorously established. Their role and limitations in relation to realistic turbulent cascades are revisited later within the two-level entropy framework.

Next, as a condition ensuring the vanishing of boundary terms, we adopt, for example,

$$\underset{\xi \to \pm \mathrm{\infty }}{\lim }J(\xi ,t)=0,$$

(26)

(or, on a finite interval, a reflecting boundary condition $J=0$). Integrating (23) over $\xi$ then gives

$${\displaystyle \frac{d}{dt}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(\xi ,t)\hspace{0.17em}d\xi =-\left[J\right(\xi ,t){]}_{\xi =-\mathrm{\infty }}^{\xi =\mathrm{\infty }}=0,$$

(27)

so that the initial normalization is preserved for all time. Within the setting (25)–(27), stationary solutions $P_s\left(\xi \right)$ can be characterized by the zero-flux condition, namely

$$J\left(\xi \right)\equiv 0.$$

(28)

A stationary distribution satisfying (28) is therefore natural as a reference distribution, and from (23), one obtains

$${\partial }_{\xi }\hspace{0.17em}\left(D\left(\xi \right)\hspace{0.17em}{P}_s\left(\xi \right)\right)=A\left(\xi \right)\hspace{0.17em}{P}_s\left(\xi \right).$$

(29)

In Section 3, we take this $P_s$ as the reference and establish the monotonicity of the relative entropy rigorously. By contrast, the “constant and nonzero energy flux” observed in the inertial subrange of turbulence corresponds intrinsically to an open system with injection and dissipation (a NESS), and thus typically permits

$$J\left(\xi \right)\equiv J_0\ne 0.$$

(30)

However, (30) is incompatible with the zero-flux boundary condition (closed-system setting) such as (26), and therefore lies outside the assumptions under which an $H$-theorem for conservative Markov semigroups is formulated. This motivates the two-level structure adopted in this paper: in Section 3, we rigorously formalize irreversibility (the arrow of time) using a conservative (closed) FP formulation, whereas from Section 4 onward we analyze the dissipation-range exponential decay as a stationary boundary-value problem for an open system using an unnormalized quantity (denoted separately in that context).

Finally, in the inertial subrange $k_f\ll k\ll k_d$, Kolmogorov scaling $E\left(k\right)\propto k^{-5/3}$ implies $kE\left(k\right)\propto k^{-2/3}$. Using the definition in Section 2.3, $P\left(\xi \right)\propto kE\left(k\right)$ (with a constant normalization factor) together with $\xi =\mathrm{l}\mathrm{n}(k/k_0)$, we obtain

$$P_{\mathrm{i}\mathrm{n}}\left(\xi \right)\propto e^{-\hspace{0.17em}{\displaystyle \frac{2}{3}}\xi }\left(k_f\ll k\ll k_d\right).$$

(31)

This provides a concrete asymptotic input to the effective theory, in the sense that the coefficients $A$ and $D$ are not arbitrary, but are uniquely determined as the first and second jump moments of the underlying coarse-grained transition kernel, and therefore fix the associated Fokker–Planck equation.

2.5. Summary of Section 2

This section formulates the spectral budget of the turbulent cascade as the first stage (the dynamical level) of the two-level theory developed in subsequent sections. The balance of $E(k,t)$ is presented through the Lin equation, confirming the inertial-subrange constant flux $\mathsf{\Pi }\left(k\right)\simeq \epsilon$ and the $-5/3$ law. A unified representation that includes the dissipation range, $E\left(k\right)=C_K{\epsilon }^{2/3}{k}^{-5/3}f(k/k_d)$, is introduced and summarized in Equations (1)–(7).

By mapping to logarithmic wavenumber space via $\xi =\mathrm{l}\mathrm{n}(k/k_0)$, we defined, with strict separation, the unnormalized density $\overline{P}(\xi ,t)=kE(k,t)$ and the normalized probability density $P(\xi ,t)=\overline{P}(\xi ,t)/E_{\mathrm{t}\mathrm{o}\mathrm{t}}\left(t\right)$ (Equations (8)–(14)). The nonlocal flux was then coarse-grained as an accumulation of near-neighbor jumps. Through the master equation, the Kramers–Moyal expansion, and the diffusion approximation, this procedure yields a conservative Fokker–Planck equation together with the conservative-form probability current $J$ (Equations (15)–(24)). To ensure mass conservation and mathematical well-posedness, the condition $D(\xi ,t)\ge d_0>0$ and appropriate boundary conditions were imposed, and the zero-flux stationary distribution $P_s$ was defined as a reference distribution (Equations (25)–(29)).

These results establish a conservative (closed-system) Fokker–Planck framework and provide the prerequisites for formulating the H-theorem (relative-entropy monotonicity) in Section 3. At the same time, physically stationary turbulence is characterized by a nonzero flux, corresponding to an open-system nonequilibrium steady state, which is incompatible with the closed-system zero-flux setting. This tension motivates the two-level separation adopted here: the conservative Fokker–Planck framework is used to establish irreversibility rigorously, while the dissipation-range exponential decay is treated separately, from Section 4 onward, as a stationary boundary-value problem for an open system (Equations (30) and (31)).

3. Fokker–Planck Description and Monotonicity of Relative Entropy (H-Theorem)

In this section, building on the normalized probability density $P(\xi ,t)$ and probability current $J(\xi ,t)$ on the logarithmic wavenumber coordinate $\xi =\mathrm{l}\mathrm{n}(k/k_0)$ introduced in Section 2, we formulate irreversibility generated by a conservative Fokker–Planck description in a rigorous manner using relative entropy (the Kullback–Leibler divergence). Since Shannon-type differential entropy does not, in general, possess a monotonicity property for diffusion processes, we take, as the fundamental quantity, the relative entropy with respect to a reference distribution $P_s\left(\xi \right)$, and compute its time evolution explicitly. The conclusion is twofold: for a closed system (under conditions that eliminate boundary contributions), the relative turbulence entropy is monotone-increasing; for an open system, one obtains a general identity that includes boundary terms.

3.1. Conservative Fokker–Planck Equation and Stationary Distribution

We recall the conservative-form evolution equation derived in Section 2:

$${\partial }_{t}P\left(\xi ,t\right)=-{\partial }_{\xi }J\left(\xi ,t\right).$$

(32)

The probability current $J$ is defined in terms of the drift coefficient $A$ and diffusion coefficient $D$ by

$$J\left(\xi ,t\right)=A\left(\xi ,t\right)P\left(\xi ,t\right)-{\partial }_{\xi }\hspace{0.17em}\left(D\left(\xi ,t\right)P\left(\xi ,t\right)\right).$$

(33)

Henceforth, to state the $H$-theorem of this section in a strict form, we assume that the coefficients are autonomous (time-independent),

$$A\left(\xi ,t\right)=A\left(\xi \right),D\left(\xi ,t\right)=D\left(\xi \right),$$

(34)

which facilitates the use of standard Markov semigroup theory. Moreover, in order to render later quadratic representations well-defined, we assume

$$D\left(\xi \right)\ge d_0>0 \left(\mathrm{all} \xi \in \mathbb{R}\right),$$

(35)

since $D\ge 0$ alone would allow division by zero at points where $D=0$. We also assume strict positivity of the density,

$$P\left(\xi ,t\right)>0 \left(\mathrm{a}.\mathrm{e}. \mathrm{in} \xi ,\forall t\right),$$

(36)

so that $\mathrm{l}\mathrm{n}(P/P_s)$ and $J^2/\left(DP\right)$ are well-defined. Normalization follows from Section 2:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(\xi ,t)\hspace{0.17em}d\xi =1$$

(37)

Differentiating in time yields,

$${\displaystyle \frac{d}{dt}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(\xi ,t\right)\hspace{0.17em}d\xi ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\partial }_{t}P(\xi ,t)\hspace{0.17em}d\xi =0.$$

(38)

A stationary state $P_s\left(\xi \right)$ satisfies ${\partial }_t{P}_s=0$, and thus, from the conservative form (32),

$${\partial }_{\xi }{J}_s\left(\xi \right)=0.$$

(39)

Hence, the stationary current is constant:

$$J_s\left(\xi \right)=J_0 \left(\mathrm{constant}\right).$$

(40)

Applying (33) in the stationary setting, $P_s$ satisfies the first-order equation

$$\left(D\right(\xi \left)P_s\right(\xi ){)}^{\prime }-A(\xi \left)P_s\right(\xi )+J_0=0.$$

(41)

As the reference distribution for a closed system (a conservative semigroup), we adopt, in this section, the zero-flux stationary distribution, i.e.,

$$J_0=0.$$

(42)

and define $P_s$ by

$$\left(D\right(\xi \left)P_s\right(\xi ){)}^{\prime }-A(\xi \left)P_s\right(\xi )=0.$$

(43)

We further impose normalization as a probability density:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{P}_s\left(\xi \right)\hspace{0.17em}d\xi =1.$$

(44)

Solving (43) for $A$ under $P_s>0$ gives

$$A\left(\xi \right)={\displaystyle \frac{{\left(D\right(\xi \left)P_s\right(\xi \left)\right)}^{\prime }}{P_s\left(\xi \right)}}.$$

(45)

Expanding the right-hand side yields

$$A\left(\xi \right)=D^{\prime }\left(\xi \right)+D\left(\xi \right){\displaystyle \frac{P_s^{\prime }\left(\xi \right)}{P_s\left(\xi \right)}}.$$

(46)

3.2. Definition of Relative Entropy

We define the relative entropy (KL divergence) of $P(\cdot ,t)$ with respect to the reference distribution $P_s$ by

$$D\hspace{0.17em}\left(P(\cdot ,t)\hspace{0.17em}\parallel \hspace{0.17em}{P}_s\right):={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(\xi ,t)\mathrm{l}\mathrm{n}\hspace{0.17em}{\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}\hspace{0.17em}d\xi .$$

(47)

In this paper, we work with a quantity that increases in time; accordingly, we define the relative turbulence entropy by

$$S_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right):=-D\hspace{0.17em}\left(P\left(\cdot ,t\right)\hspace{0.17em}\parallel \hspace{0.17em}{P}_s\right)=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P\left(\xi ,t\right)\ln \hspace{0.17em}{\displaystyle \frac{P\left(\xi ,t\right)}{P_s\left(\xi \right)}}\hspace{0.17em}d\xi .$$

(48)

The Kullback–Leibler divergence was originally introduced in information theory as a measure of statistical distinguishability between probability distributions [31,32]. It has since become a central quantity in nonequilibrium statistical mechanics and large-deviation theory, where it plays a fundamental role in the formulation of entropy production and irreversibility [33].

3.3. H-Theorem and a General Identity for Open Systems

This subsection establishes an $H$-theorem for the conservative Fokker–Planck dynamics introduced in Section 2.4. Under the assumptions of mass conservation, strictly positive diffusion, and vanishing probability flux at the boundary, we show that the relative entropy with respect to the stationary reference distribution is a Lyapunov functional. The relative turbulence entropy therefore increases monotonically in time, providing a rigorous formulation of irreversibility at the dynamical level.

This result places the coarse-grained cascade dynamics within the general framework of conservative Markov semigroups and clarifies the thermodynamic structure underlying the evolution in logarithmic wavenumber space. The $H$-theorem derived here constitutes the first level of the two-level framework proposed in this paper, furnishing the dynamical arrow of time that will later be complemented, in Section 4 and Section 5, by a variational selection principle for the dissipation-range asymptotics.

3.3.1. Gradient Form of the Probability Current (Consequence of the Zero-Flux Reference Distribution)

Substituting the expression of the drift coefficient (46), into the definition of the probability current (33), and collecting terms, one finds after straightforward algebra that

$$J(\xi ,t)=-D\left(\xi \right)\hspace{0.17em}\left(P^{\prime }(\xi ,t)-{\displaystyle \frac{P_s^{\prime }\left(\xi \right)}{P_s\left(\xi \right)}}\hspace{0.17em}P(\xi ,t)\right).$$

(49)

The expression in parentheses coincides with the derivative of the ratio $P/P_s$, since

$${\partial }_{\xi }\hspace{0.17em}\left({\displaystyle \frac{P}{P_s}}\right)={\displaystyle \frac{1}{P_s}}\hspace{0.17em}\left(P^{\prime }−{\displaystyle \frac{P_s^{\prime }}{P_s}}P\right).$$

(50)

Therefore, the probability current can be written in the gradient form

$$J\left(\xi ,t\right)=-D\left(\xi \right)P_s\left(\xi \right)\hspace{0.17em}{\partial }_{\xi }\hspace{0.17em}\left({\displaystyle \frac{P\left(\xi ,t\right)}{P_s\left(\xi \right)}}\right).$$

(51)

Equivalently, using

$${\partial }_{\xi }\mathrm{l}\mathrm{n}\left({\displaystyle \frac{P}{P_s}}\right)={\displaystyle \frac{1}{P}}\hspace{0.17em}\left(P^{\prime }−{\displaystyle \frac{P_s^{\prime }}{P_s}}P\right),$$

(52)

the current admits the alternative representation

$$J(\xi ,t)=-D\left(\xi \right)P(\xi ,t)\hspace{0.17em}{\partial }_{\xi }\left[ln\left({\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}\right)\right]\hspace{0.17em}.$$

(53)

In what follows, we refer to either (51) or (53) as the gradient-form representation of the probability current associated with the zero-flux stationary reference distribution.

3.3.2. Time Derivative of the Relative Entropy

We differentiate (48) in time. Formally,

$${\displaystyle \frac{d}{dt}}D\left(P\parallel P_s\right)={\displaystyle \frac{d}{dt}}\int P\left(ln{\displaystyle \frac{P}{P_s}}\right)\hspace{0.17em}d\xi .$$

(54)

Assuming that the exchange of differentiation and integration is justified (under integrability and regularity assumptions),

$${\displaystyle \frac{d}{dt}}D(P\parallel P_s)=\int {\partial }_{t}P\hspace{0.17em}\mathrm{l}\mathrm{n}\hspace{0.17em}{\displaystyle \frac{P}{P_s}}\hspace{0.17em}d\xi +\int P\hspace{0.17em}{\partial }_t\hspace{0.17em}\left(ln{\displaystyle \frac{P}{P_s}}\right)d\xi .$$

(55)

Since $P_s$ is stationary and independent of $t$,

$${\partial }_t\hspace{0.17em}\hspace{0.17em}\left(ln{\displaystyle \frac{P}{P_s}}\right)={\partial }_t\left(lnP\right)={\displaystyle \frac{{\partial }_{t}P}{P}}.$$

(56)

Therefore, the second term becomes

$$\int P\hspace{0.17em}{\partial }_t\hspace{0.17em}\hspace{0.17em}\left(ln{\displaystyle \frac{P}{P_s}}\right)d\xi =\int P\hspace{0.17em}{\displaystyle \frac{{\partial }_{t}P}{P}}\hspace{0.17em}d\xi =\int {\partial }_{t}P\hspace{0.17em}d\xi .$$

(57)

By (41), $\int {\partial }_{t}P\hspace{0.17em}d\xi =0$, and thus (55) reduces to

$${\displaystyle \frac{d}{dt}}D(P\parallel P_s)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\partial }_{t}P(\xi ,t)\hspace{0.17em}\left(ln{\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}\right)\hspace{0.17em}d\xi$$

(58)

Substituting (32) yields

$${\displaystyle \frac{d}{dt}}D\left(P\parallel P_s\right)=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\partial }_{\xi }J\left(\xi ,t\right)\left(\hspace{0.17em}ln{\displaystyle \frac{P\left(\xi ,t\right)}{P_s\left(\xi \right)}}\hspace{0.17em}\right)d\xi .$$

(59)

3.3.3. Integration by Parts and a Quadratic Dissipation Representation

Integrating of (59) by parts and keeping boundary contributions explicit, we obtain

$${\displaystyle \frac{d}{dt}}D(P\parallel P_s)=-[\hspace{0.17em}J(\xi ,t)\hspace{0.17em}\mathrm{l}\mathrm{n}({\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}){]}_{-\mathrm{\infty }}^{\mathrm{\infty }}+{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}J(\xi ,t)\hspace{0.17em}{\partial }_{\xi }\mathrm{l}\mathrm{n}({\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}})\hspace{0.17em}d\xi .$$

(60)

Substituting (53) into the bulk term of (60), the integrand becomes a nonnegative quadratic form in the current. After this straightforward substitution, one arrives at

$${\displaystyle \frac{d}{dt}}D(P\parallel P_s)=-[\hspace{0.17em}J(\xi ,t)\hspace{0.17em}\mathrm{l}\mathrm{n}({\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}){]}_{-\mathrm{\infty }}^{\mathrm{\infty }}-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{J(\xi ,t{)}^2}{D\left(\xi \right)P(\xi ,t)}}\hspace{0.17em}d\xi .$$

(61)

Equation (61) is the fundamental dissipation identity for the relative entropy. The second term on the right-hand side is manifestly nonpositive under the assumptions $D\left(\xi \right)>0$ and $P(\xi ,t)>0$, while the first term represents possible boundary contributions. This structure directly leads to the $H$-theorem for closed systems and to a generalized entropy balance for open systems, as discussed below.

3.3.4. H-Theorem for a Closed System

For a closed system, i.e., a conservative Markov semigroup, we assume boundary conditions such that the boundary contribution in the entropy balance vanishes. Under this assumption, the boundary term in the dissipation identity (61) is zero,

$${\left[\hspace{0.17em}J(\xi ,t)\hspace{0.17em}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}\right)\right]}_{-\infty }^{\infty }=0.$$

(62)

Consequently, (61) reduces to

$${\displaystyle \frac{d}{dt}}D(P\parallel P_s)=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{J(\xi ,t{)}^2}{D\left(\xi \right)P(\xi ,t)}}\hspace{0.17em}d\xi \hspace{0.25em} \le \hspace{0.25em} 0,$$

(63)

where the inequality follows from the strict positivity of $D\left(\xi \right)$ and $P(\xi ,t)$. Recalling that the relative turbulence entropy is defined by $S_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)=-D(P\parallel P_s)$, we equivalently obtain

$${\displaystyle \frac{d}{dt}}{S}_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{J(\xi ,t{)}^2}{D\left(\xi \right)P(\xi ,t)}}\hspace{0.17em}d\xi \hspace{0.25em} \ge \hspace{0.25em} 0.$$

(64)

Equations (63) and (64) constitute the $H$-theorem for the conservative Fokker–Planck description: the relative turbulence entropy $S_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)$ is a Lyapunov functional and increases monotonically in time.

3.3.5. Open Systems: General Identity with Boundary Terms

For an open system that permits injection and outflow through the boundary, (62) need not hold in general, and (61) remains as a general identity. Namely,

$${\displaystyle \frac{d}{dt}}D(P\parallel P_s)=-\sigma \left(t\right)-[J\left(\xi ,t\right)\hspace{0.17em}\ln \hspace{0.17em}{\displaystyle \frac{P\left(\xi ,t\right)}{P_s\left(\xi \right)}}{]}_{-\mathrm{\infty }}^{\mathrm{\infty }}.$$

(65)

Equivalently, using $S_{\mathrm{r}\mathrm{e}\mathrm{l}}=-D$,

$${\displaystyle \frac{d}{dt}}{S}_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)=\sigma \left(t\right)+\left[J\right(\xi ,t)\hspace{0.17em}\mathrm{l}\mathrm{n}{\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}{]}_{-\mathrm{\infty }}^{\mathrm{\infty }}$$

(66)

The second term on the right-hand side represents the inflow/outflow of information (relative entropy) through the boundary, and it can break monotonicity. Such boundary contributions and their role in entropy balance are a generic feature of stochastic thermodynamics in open nonequilibrium systems [34]. Therefore, for the physically stationary turbulent state (a NESS) in which a nonzero flux is essential, it is not appropriate to apply the closed-system monotonicity statement (64) verbatim; rather, it is more suitable to organize the contributions in the form (65) and (66).

3.4. Summary of Section 3

Under the conservative-form Fokker–Planck description introduced in Section 2 (Equations (32) and (33)), choosing the zero-flux stationary distribution $P_s$ (Equations (43) and (44)) as the reference distribution yields a gradient-form representation of the probability current (Equations (51)–(53)). This structure leads directly to an H-theorem stating that the relative turbulence entropy $S_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)$ increases monotonically in a closed system (Equation (64)).

For open systems, the result is naturally expressed as a general identity that includes boundary terms (Equations (65) and (66)), and monotonicity may be violated depending on the boundary conditions. In the next section, the dissipation range is therefore treated outside the conservative setting, as a stationary boundary-value problem with killing (absorption), and constraints on exponential decay are derived using a WKB approximation.

The monotonicity result obtained here is an instance of the classical H-theorem for Markovian dynamics and is consistent with general entropy-production theory in nonequilibrium statistical mechanics [29,35,36].

4. Asymptotic Structure of the Dissipation-Range Spectrum via WKB Analysis

In the preceding sections, we established a conservative Fokker–Planck (FP) framework to describe the inertial-range energy spectrum. By treating the turbulence cascade as a probability-preserving process in logarithmic wavenumber space, we showed that the spectral form is constrained by an H-theorem rooted in relative-entropy monotonicity. This formalization successfully captures the essence of the cascade dynamics—where energy is redistributed across scales without net loss—and provides a rigorous basis for the emergence of inertial-range scaling laws.

However, the dissipation range lies fundamentally beyond the reach of such conservative descriptions. At sufficiently large wavenumbers, the dominance of viscous effects introduces a localized, irreversible sink of energy. This “leakage” of energy from the scale-space continuum violates the conservation of probability required for standard semigroup operators. Consequently, the selection of the spectral shape can no longer be dictated solely by entropy-based variational principles. Instead, the dissipation-range profile must emerge from a structural modification of the underlying transport equation.

The objective of this section is to reformulate the cascade dynamics by incorporating a “Killing term” to account for viscous absorption. We show how the dissipation-range tail is uniquely selected through a dynamical competition between cascade-driven diffusion and viscous destruction.

4.1. Formulation: Stationary Fokker–Planck Equation with Killing

We take, as a starting point, the conservative Fokker–Planck description established in Section 2 and Section 3, formulated in terms of the logarithmic wavenumber variable $\xi =\mathrm{l}\mathrm{n}(k/k_0)$, the normalized density $P(\xi ,t)$, and the associated probability current $J(\xi ,t)$. To account for viscous dissipation in the high-wavenumber regime, we introduce a killing (absorption) term that explicitly removes probability from scale space.

The stationary Fokker–Planck equation with killing is written as

$$-{\partial }_{\xi }\left(A\right(\xi )\hspace{0.17em}\stackrel{\sim }{P}(\xi \left)\right)+{\partial }_{\xi }^2\left(D\right(\xi )\hspace{0.17em}\stackrel{\sim }{P}(\xi \left)\right)-\mathsf{\Sigma }\left(\xi \right)\hspace{0.17em}\stackrel{\sim }{P}\left(\xi \right)=0,$$

(67)

where $\stackrel{\sim }{P}\left(\xi \right)$ denotes the unnormalized density related to the energy spectrum by $\stackrel{\sim }{P}\left(\xi \right)=kE\left(k\right)$. The functions $A\left(\xi \right)$ and $D\left(\xi \right)$ represent, respectively, the effective drift and diffusion generated by the coarse-grained cascade dynamics, while $\mathsf{\Sigma }\left(\xi \right)\ge 0$ denotes the killing rate associated with viscous dissipation.

Equation (67) can, equivalently, be written in flux form as

$${\partial }_{\xi }J\left(\xi \right)=-\mathsf{\Sigma }\left(\xi \right)\hspace{0.17em}\stackrel{\sim }{P}\left(\xi \right),J\left(\xi \right)=A\left(\xi \right)\stackrel{\sim }{P}\left(\xi \right)-{\partial }_{\xi }\left(D\right(\xi \left)\stackrel{\sim }{P}\right(\xi \left)\right),$$

(68)

which makes explicit that probability flux in logarithmic wavenumber space is no longer conserved. Instead, probability is continuously absorbed at small scales, reflecting the irreversible removal of energy by viscosity. The detailed derivation of the stationary formulation (67) and (68) and its equivalent second-order representation is provided in Appendix B.1.

4.2. Boundary-Value Formulation and Asymptotic Regime

The dissipation range corresponds to the limit $\xi \to +\mathrm{\infty }$, where the stationary solution of Equation (67) must decay sufficiently rapidly to ensure integrability of the energy spectrum. This requirement is expressed as

$$\stackrel{\sim }{P}\left(\xi \right)\to 0(\xi \to +\mathrm{\infty }).$$

(69)

Because probability is not conserved in the presence of killing, Equation (67) does not admit an H-theorem analogous to that of the conservative Fokker–Planck equation. The dissipation-range spectrum must therefore be selected dynamically as a stationary solution of a boundary-value problem rather than as the outcome of entropy maximization.

For subsequent analysis, it is convenient to rewrite Equation (67) in the equivalent second-order form

$$\left(D\stackrel{\sim }{P}{)}^{″}-(A\stackrel{\sim }{P}{)}^{\prime }-\mathsf{\Sigma }\stackrel{\sim }{P}=0.\right.$$

(70)

This representation highlights the competition between diffusion, drift, and absorption, and serves as the starting point for the asymptotic analysis of the dissipation range.

We assume that, at sufficiently large $\xi$, the coefficients $A\left(\xi \right)$, $D\left(\xi \right)$, and $\mathsf{\Sigma }\left(\xi \right)$ vary slowly compared with the rapid decay of $\stackrel{\sim }{P}\left(\xi \right)$. This scale separation is the essential condition that permits a WKB (Liouville–Green) treatment of Equation (70). The boundary-value formulation (69) and (70) and its reformulation in terms of a second-order differential equation are derived explicitly in Appendix B.1.

4.3. WKB Structure and Dominant Balance

To extract the leading asymptotic behavior of $\stackrel{\sim }{P}\left(\xi \right)$, we introduce the WKB ansatz

$$\stackrel{\sim }{P}\left(\xi \right)=Q\left(\xi \right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{1}{\delta }}S\left(\xi \right)\right],0<\delta \ll 1,$$

(71)

where the function $S\left(\xi \right)$ governs the dominant exponential decay and $Q\left(\xi \right)$ varies slowly in comparison. The substitution of Equation (71) into Equation (70), followed by ordering in powers of the small parameter $\delta$, yields at leading order the eikonal equation

$$D\left(\xi \right)\left(S^{\prime }\right(\xi ){)}^2=\mathsf{\Sigma }(\xi ).$$

(72)

This equation identifies the dominant balance controlling the dissipation range. The exponential decay of the spectrum is determined by the competition between the effective diffusion coefficient $D\left(\xi \right)$, which encapsulates the spreading of energy in scale space induced by the cascade, and the killing rate $\mathsf{\Sigma }\left(\xi \right)$, which represents irreversible viscous absorption. Drift effects associated with $A\left(\xi \right)$ enter only at subleading order and therefore do not affect the leading decay exponent.

The physically relevant solution of Equation (72) is obtained by selecting the positive root,

$$S^{\prime }\left(\xi \right)=\sqrt{{\displaystyle \frac{\mathsf{\Sigma }\left(\xi \right)}{D\left(\xi \right)}}},$$

(73)

which ensures decay of $\stackrel{\sim }{P}\left(\xi \right)$ as $\xi \to +\mathrm{\infty }$. The WKB substitution (71) and the ordering procedure leading to the eikonal Equation (72) are developed in detail in Appendix B.2.

4.4. Asymptotic Scaling in the Dissipation Range

In the viscous regime, the killing rate is determined by the molecular viscosity and scales as

$$\mathsf{\Sigma }\left(\xi \right)\sim 2\nu k^2\propto e^{2\xi },$$

(74)

reflecting the quadratic dependence of viscous dissipation on the wavenumber. In contrast, the diffusion coefficient associated with the cascade is assumed to decay algebraically in logarithmic wavenumber space,

$$D\left(\xi \right)\sim D_0{e}^{-\gamma \xi },\gamma >0.$$

(75)

Substituting Equations (74) and (75) into Equation (73) yields

$$S^{\prime }\left(\xi \right)\sim e^{(2+\gamma )\xi /2},$$

(76)

which, upon integration, gives

$$S\left(\xi \right)\sim e^{(2+\gamma )\xi /2}.$$

(77)

This result shows that the exponential decay of the dissipation-range spectrum originates from the mismatch between the rapid growth of viscous absorption and the decay of cascade-induced diffusion at large wavenumbers. The asymptotic evaluation based on the scaling assumptions (74) and (75) and the resulting expressions (76) and (77) are obtained in Appendix B.3.

4.5. Dissipation-Range Energy Spectrum

Using the relation $\xi =\mathrm{l}\mathrm{n}(k/k_0)$, the exponential factor in Equation (71) can be expressed in wavenumber space as

$$\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{1}{\delta }}S\left(\xi \right)\right]\sim \mathrm{e}\mathrm{x}\mathrm{p}\left[-{\beta }_m{\left({\displaystyle \frac{k}{k_d}}\right)}^{\hspace{0.17em}1+\gamma /2}\right],$$

(78)

where ${\beta }_m>0$ absorbs numerical constants and matching coefficients.

Since $\stackrel{\sim }{P}\left(\xi \right)=kE\left(k\right)$, the energy spectrum behaves as

$$E\left(k\right)\sim k^{-1}Q\left(\mathrm{l}\mathrm{n}k\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\beta }_m{\left({\displaystyle \frac{k}{k_d}}\right)}^{\hspace{0.17em}1+\gamma /2}\right].$$

(79)

Matching this asymptotic form to the inertial-range Kolmogorov spectrum yields the final dissipation-range expression

$$E\left(k\right)\sim C_K\hspace{0.17em}{\epsilon }^{{\displaystyle \frac{2}{3}}}{k}^{-{\displaystyle \frac{5}{3}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\beta }_m{\left({\displaystyle \frac{k}{k_d}}\right)}^m\right], m=1+{\displaystyle \frac{\gamma }{2}}.$$

(80)

The transformation from logarithmic wavenumber space to physical wavenumber space, as well as the matching leading to the dissipation-range spectrum (78)–(80), are detailed in Appendix B.4.

4.6. Interpretation of the Exponent Selection

Equation (80) shows that the dissipation-range exponent $m$ is not an adjustable fitting parameter. Instead, it is uniquely fixed by the asymptotic scaling of the diffusion coefficient $D\left(\xi \right)$, which encodes the effective nonlocality of the cascade dynamics in logarithmic scale space. The stretched-exponential cutoff therefore emerges as a direct consequence of the interplay between cascade-induced diffusion and viscous killing, rather than as an empirical modification imposed on the inertial-range spectrum. The mathematical structure underlying the exponent selection, arising from the WKB order-by-order decomposition, is summarized in Appendix B.4.

4.7. Summary of the Dissipation-Range Selection Mechanism

The analysis presented in this section leads to three central conclusions. First, the dissipation range must be described by a nonconservative Fokker–Planck equation with killing, reflecting the irreversible removal of energy by viscosity. Second, the leading asymptotic behavior of the stationary solution is governed by a WKB eikonal balance between diffusion and killing. Third, this balance uniquely determines a Kolmogorov spectrum with a stretched-exponential cutoff, with exponent

$$m=1+{\displaystyle \frac{\gamma }{2}}.$$

(81)

All intermediate derivations and subleading corrections are deferred to Appendix B, where the WKB hierarchy is developed in full detail.

5. Two-Level Entropy Principle and Determination of the Dissipation-Tail Exponent

In Section 4, the dissipation-range asymptotics were derived by formulating a stationary boundary-value problem for the Fokker–Planck equation with killing and analyzing its high-wavenumber limit using WKB methods. The central result is that the dissipation-range tail belongs to a stretched-exponential family, with an exponent governed by the mismatch between the viscous killing rate and the decay of the effective diffusion coefficient.

This section incorporates that asymptotic result into a closed predictive framework by distinguishing two logically separate roles. The first is the dynamical level, which ensures the irreversibility of time evolution through an autonomous conservative Fokker–Planck semigroup. The second is the selection level, which determines the stationary spectral shape—including the dissipation-range cutoff—via an information-theoretic principle. Detailed derivations omitted from the main text are provided in Appendix C, and each subsection below indicates the relevant part of Appendix C for reference.

5.1. Dynamical Level: Autonomous Conservative FP Semigroup and the H-Theorem

We use the logarithmic wavenumber variable introduced in Section 2,

$$\xi =\mathrm{l}\mathrm{n}(k/k_0),$$

(82)

together with a normalized probability density $P(\xi ,t)$ satisfying

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(\xi ,t)\hspace{0.17em}d\xi =1.$$

(83)

In this section, we consider an autonomous (time-independent) conservative Fokker–Planck operator, meaning that the drift and diffusion coefficients $A\left(\xi \right)$ and $D\left(\xi \right)$ are independent of time. The evolution of $P(\cdot ,t)$ is therefore governed by an autonomous Markov semigroup $T_t=e^{tL}$. This autonomy is essential: it makes it meaningful to speak of a stationary reference distribution associated with the operator itself, rather than with a particular transient state.

The conservative Fokker–Planck equation is written in flux form as

$${\partial }_{t}P(\xi ,t)=-{\partial }_{\xi }J(\xi ,t),$$

(84)

with probability flux

$$J(\xi ,t)=A\left(\xi \right)P(\xi ,t)-{\partial }_{\xi }\left(D\right(\xi \left)P\right(\xi ,t\left)\right).$$

(85)

For the H-theorem we assume uniform positivity of the diffusion coefficient,

$$D\left(\xi \right)\ge d_0>0,$$

(86)

which is not a cosmetic technicality: it guarantees that the dissipation density $J^2/\left(DP\right)$ is well-defined and prevents degenerate “deterministic” limits in which scale-space transport loses its irreducible stochasticity. We also impose boundary conditions (reflecting boundaries or sufficiently rapid decay at infinity) under which boundary contributions vanish; the detailed integration-by-parts steps are given in Appendix C.1.

We define the relative entropy (Kullback–Leibler divergence) of $P(\cdot ,t)$ with respect to a stationary reference distribution $P_s$ by

$$D\left(P\right(\cdot ,t)\hspace{0.17em}\parallel \hspace{0.17em}{P}_s)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(\xi ,t)\mathrm{l}\mathrm{n}{\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}\hspace{0.17em}d\xi ,$$

(87)

and—using the stationary balance relation implied by zero flux—the flux admits the gradient form

$$J(\xi ,t)=-D\left(\xi \right)P(\xi ,t)\hspace{0.17em}{\partial }_{\xi }\hspace{0.17em}\left[\ln {\displaystyle \frac{P(\xi ,t)}{P_s\left(\xi \right)}}\right].$$

(88)

Substituting this structure into the time derivative of the KL divergence yields the quadratic dissipation form

$${\displaystyle \frac{d}{dt}}D(P\hspace{0.17em}\parallel \hspace{0.17em}{P}_s)=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{J(\xi ,t{)}^2}{D\left(\xi \right)P(\xi ,t)}}\hspace{0.17em}d\xi \le 0,$$

(89)

equivalently,

$${\displaystyle \frac{d}{dt}}{S}_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{J(\xi ,t{)}^2}{D\left(\xi \right)P(\xi ,t)}}\hspace{0.17em}d\xi \ge 0,$$

(90)

where $S_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(t\right)\equiv -\hspace{0.17em}D\left(P\right(\cdot ,t)\hspace{0.17em}\parallel \hspace{0.17em}{P}_s)$. This is the dynamical-level irreversibility statement: the semigroup possesses a Lyapunov functional, hence an “arrow of time,” purely from conservative scale-space transport. However—and this is the key limitation—the H-theorem is tied to a zero-flux reference distribution. It therefore does not determine the spectral shape of a nonequilibrium steady cascade with nonzero flux, and in particular, it does not fix the dissipation-tail exponent $m$. Determining $m$ requires an additional principle beyond conservative dynamics. The construction of $P_s$ and the derivation of (88)–(90) are detailed in Appendix C.1.

5.2. Consequence of the WKB Analysis and the Stretched-Exponential Tail

To describe the dissipation-range shape itself, we work with the unnormalized density

$$\overline{P}(\xi ,t)\equiv k\hspace{0.17em}E(k,t)(k=k_0{e}^{\xi }),$$

(91)

which directly represents the energy distribution in logarithmic wavenumber space. In the dissipation range, we adopted an effective equation with killing:

$${\partial }_t\overline{P}=-{\partial }_{\xi }\hspace{0.17em}\left(A\left(\xi \right)\overline{P}-{\partial }_{\xi }\left(D\right(\xi \left)\overline{P}\right)\right)-\mathsf{\Sigma }\left(\xi \right)\overline{P}+G(\xi ,t),$$

(92)

and assumed that injection is negligible at sufficiently large $\xi$,

$$G\left(\xi ,t\right)\simeq 0 \left(\xi \mathrm{sufficiently} \mathrm{large}\right),$$

(93)

together with stationarity ${\partial }_t\overline{P}=0$. Under these assumptions, the dissipation-range shape is governed by the stationary boundary-value problem

$$\left(D\right(\xi \left)\overline{P}\right(\xi ){)}^{″}-(A\left(\xi \right)\overline{P}\left(\xi \right){)}^{\prime }-\mathsf{\Sigma }\left(\xi \right)\overline{P}\left(\xi \right)=0.$$

(94)

The physical content of (94) is clear: the dissipation range is not described by probability conservation but by a balance between (i) scale-space transport encoded in $A, D$ and (ii) absorption encoded in $\mathsf{\Sigma }$, with the boundary condition that $\overline{P}\left(\xi \right)\to 0$ (and the corresponding flux tends to zero) as $\xi \to +\mathrm{\infty }$. The WKB analysis in Section 4 shows that, under sufficient smoothness and slow variation of coefficients and in the regime where the exponential phase dominates, the decaying solution possesses a stretched-exponential tail:

$$E\left(k\right)\sim C_K\hspace{0.17em}{\epsilon }^{{\displaystyle \frac{2}{3}}}{k}^{-{\displaystyle \frac{5}{3}}}\exp \left[-\beta (k/k_d{)}^m\right] (k\to \mathrm{\infty }),$$

(95)

where the Kolmogorov wavenumber is

$$k_d=(\epsilon /{\nu }^3{)}^{1/4}.$$

(96)

Within the present effective-theory viewpoint, the “rigor” is conditional: given (92)–(94) and the WKB assumptions, the dissipation tail reduces to the family (95). However, neither $\beta$ nor $m$ is fixed by local asymptotics alone; selecting them requires global information and an additional selection principle. Using the WKB results derived in Section 4 and Appendix B as input, Appendix C.2 formulates the dissipation-range problem as a stationary boundary-value problem at the selection level and derives the admissible stretched-exponential family (95).

5.3. Formulation of the Two-Level Principle and Parameter Identification

The two levels can be summarized as follows: the dynamical level provides a robust monotonicity principle (H-theorem), while the dissipation-range analysis constrains the tail to a stretched-exponential family. Accordingly, we introduce the candidate spectral family

$$E_{m,\beta }\left(k\right)=C_K\hspace{0.17em}{\epsilon }^{{\displaystyle \frac{2}{3}}}{k}^{-{\displaystyle \frac{5}{3}}}\exp \left[-\beta (k/k_d{)}^m\right], \left(\beta >0, m>0\right)$$

(97)

We restrict attention to the physically meaningful domain $\left.\mathsf{\Omega }=[{\xi }_{-},\mathrm{\infty }\right)$, where ${\xi }_{-}$ corresponds to a representative forcing wavenumber $k_f$:

$${\xi }_{-}=\ln \left({\displaystyle \frac{k_f}{k_0}}\right),$$

(98)

$$\left.\mathsf{\Omega }=[\hspace{0.17em}{\xi }_{-},\hspace{0.17em}\mathrm{\infty }\right).$$

(99)

This cutoff has a concrete meaning: it excludes the energy-containing range dominated by forcing and non-universal large-scale structures, which the present inertial-to-dissipation framework does not attempt to model. On $\mathsf{\Omega }$, we define the normalized density associated with (97) by

$$P_{m,\beta }(\xi )\equiv {\displaystyle \frac{{\overline{P}}_{m,\beta }(\xi )}{{\int }_{{\xi }_{-}}^{\mathrm{\infty }}{\overline{P}}_{m,\beta }(\eta )\hspace{0.17em}d\eta }}(\xi \in [{\xi }_{-},\mathrm{\infty })),$$

(100)

and adopt, as the selection principle, minimization of the KL divergence against a reference distribution $P_s^{\left(\gamma \right)}$:

$$D_{\mathrm{K}\mathrm{L}}(m,\beta ;\gamma )\equiv {\int }_{{\xi }_{-}}^{\mathrm{\infty }}{P}_{m,\beta }(\xi )\mathrm{l}\mathrm{n}{\displaystyle \frac{P_{m,\beta }(\xi )}{P_s^{\left(\gamma \right)}(\xi )}}\hspace{0.17em}d\xi .$$

(101)

The WKB result of Section 4 implies a coupling between the dissipation-range decay of the diffusion coefficient and the tail exponent: when $D\left(\xi \right)\sim D_{\mathrm{\infty }}{e}^{-\gamma \xi }$ as $\xi \to \mathrm{\infty }$, one obtains

$$m=1+{\displaystyle \frac{\gamma }{2}}.$$

(102)

Therefore, $m$ is not optimized independently; instead, we set

$$m\left(\gamma \right)\equiv 1+{\displaystyle \frac{\gamma }{2}},$$

(103)

reducing the selection problem to a one-parameter identification in $\gamma$. The construction of ${\overline{P}}_{m,\beta }$ and the reduction to a one-dimensional selection problem are detailed in Appendix C.3.

5.4. Construction of the Reference Distribution and a Global Model

The reference distribution $P_s^{\left(\gamma \right)}$ is defined as the zero-flux stationary distribution associated with an autonomous conservative FP operator. Its general integral form is

$$P_s\left(\xi \right)={\displaystyle \frac{C_s}{D\left(\xi \right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\int }^{\xi }{\displaystyle \frac{A\left(\eta \right)}{D\left(\eta \right)}}\hspace{0.17em}d\eta \right),$$

(104)

with the normalization constant fixed on $\left.\mathsf{\Omega }=[{\xi }_{-},\mathrm{\infty }\right)$. To use (104) in practice while respecting both inertial-to-dissipation behavior and integrability, we adopt global analytic models $A^{\left(\gamma \right)}\left(\xi \right)$ and $D^{\left(\gamma \right)}\left(\xi \right)$ that enforce: slow variation at low-to-intermediate $\xi$, the high-$\xi$ dissipation scaling $D\left(\xi \right)\sim D_{\mathrm{\infty }}{e}^{-\gamma \xi }$, and integrability of $P_s^{\left(\gamma \right)}$. Introduce a transition-width parameter $a>0$ and dissipation-entry point ${\xi }_d$, and define

$$H(\xi )\equiv {\displaystyle \frac{1}{1+e^{-a(\xi -{\xi }_d)}}},$$

(105)

then define the diffusion model

$$D^{\left(\gamma \right)}(\xi )\equiv D_0\hspace{0.17em}{e}^{{\displaystyle \frac{2}{3}}\xi }{\left(1+e^{a(\xi -{\xi }_d)}\right)}^{-{\displaystyle \frac{2/3+\gamma }{a}}}(D_0>0),$$

(106)

so that, in particular,

$$D^{\left(\gamma \right)}\left(\xi \right)\sim D_{\mathrm{\infty }}{e}^{-\gamma \xi },D_{\mathrm{\infty }}\equiv D_0\hspace{0.17em}{e}^{(2/3+\gamma ){\xi }_d}(\xi \gg {\xi }_d).$$

(107)

Next, define the drift by

$$A^{\left(\gamma \right)}\left(\xi \right)\equiv -\alpha \hspace{0.17em}{D}^{\left(\gamma \right)}\left(\xi \right)\hspace{0.17em}H\left(\xi \right)(\alpha >0),$$

(108)

which yields an explicit global reference distribution

$$P_s^{\left(\gamma \right)}\left(\xi \right)={\displaystyle \frac{C_s^{\left(\gamma \right)}}{D_0}}\hspace{0.17em}{e}^{-{\displaystyle \frac{2}{3}}\xi }{\left(1+e^{a(\xi -{\xi }_d)}\right)}^{{\displaystyle \frac{2/3+\gamma -\alpha }{a}}}.$$

(109)

The high-wavenumber decay of (109) is governed by the sign of $\gamma -\alpha$; a sufficient condition for integrability on $\mathsf{\Omega }$ is

$$\alpha >\gamma .$$

(110)

This construction makes the KL functional (101) well-defined and explicitly computable while preserving the dissipation-side transport relaxation encoded by $\gamma$. The derivation from the zero-flux condition to (104) and the detailed algebra leading to (109) and (110) are given in Appendix C.4.

5.5. Fixing Parameters by Dissipation-Rate Consistency

To avoid arbitrariness associated with keeping $\beta$ as a free parameter, we fix $\beta$ by enforcing dissipation-rate consistency. We define, with the same cutoff $k_f$,

$$\epsilon =2\nu {\int }_{k_f}^{\mathrm{\infty }}{k}^2\hspace{0.17em}{E}_{m,\beta }\left(k\right)\hspace{0.17em}dk.$$

(111)

Substituting the family (97) and performing standard changes of variables yields the implicit condition

$$\epsilon =2\nu \hspace{0.17em}{C}_K\hspace{0.17em}{\epsilon }^{2/3}\hspace{0.17em}{k}_d^{4/3}\cdot {\displaystyle \frac{1}{m}}\hspace{0.17em}{\beta }^{-4/\left(3m\right)}\hspace{0.17em}\mathsf{\Gamma }\hspace{0.17em}\left({\displaystyle \frac{4}{3m}},\beta x_f^m\right),$$

(112)

hence we define $\beta \left(m\right)$ as the unique $\beta >0$ satisfying (112),

$$\mathrm{For} \mathrm{each} m>0, \mathrm{define} \beta \left(m\right)>0 as a \left(positive\right) solution of Equation \left(112\right).$$

(113)

Using the coupling $m=m\left(\gamma \right)$ in (103), $\beta$ becomes a function of $\gamma$ alone:

$$\beta \left(\gamma \right)\equiv \beta \left(m\right(\gamma \left)\right).$$

(114)

Thus, after dissipation-rate consistency, the candidate family is parameterized by $\gamma$ only. The full derivation of (112) and the uniqueness/implicitness discussion for $\beta$ are provided in Appendix C.5.

5.6. Optimization of $\gamma$ and the Admissible Range of the Tail Exponent

Accordingly, the candidate family is fully parameterized by $\gamma$ via $m\left(\gamma \right)$ and $\beta \left(\gamma \right)$, and we define the objective functional

$$D\left(\gamma \right)\equiv {\int }_{{\xi }_{-}}^{\mathrm{\infty }}{P}_{\gamma }\left(\xi \right)\mathrm{l}\mathrm{n}{\displaystyle \frac{P_{\gamma }\left(\xi \right)}{P_s^{\left(\gamma \right)}\left(\xi \right)}}\hspace{0.17em}d\xi .$$

(115)

The admissible set $\mathsf{\Gamma }$ is determined by the (i) positivity of $D^{\left(\gamma \right)}$ (guaranteed by construction), (ii) integrability of $P_s^{\left(\gamma \right)}$ (ensured by $\alpha >\gamma$), and (iii) self-consistency of the dissipation-range diffusion relaxation and slow-variation requirements. We therefore set

$$\mathsf{\Gamma }=\left(\mathrm{0,1}\right].$$

(116)

The final identification is the one-dimensional minimization

$${\gamma }^{\ast }\in \mathrm{a}\mathrm{r}\mathrm{g} \underset{\gamma \in \mathsf{\Gamma }}{\mathrm{m}\mathrm{i}\mathrm{n}} D\left(\gamma \right),$$

(117)

which determines

$$m^{\ast }=m\left({\gamma }^{\ast }\right)=1+{\displaystyle \frac{{\gamma }^{\ast }}{2}},{ \beta }^{\ast }=\beta \left({\gamma }^{\ast }\right).$$

(118)

Moreover, combining the coupling $m=1+\gamma /2$ with the testable admissible range $\gamma \in [2/\mathrm{3,1}]$ yields

$$m\in \left[\hspace{0.17em}{\displaystyle \frac{4}{3}},{\displaystyle \frac{3}{2}}\right].$$

(119)

The WKB-side derivation of the coupling and the detailed definitions of $P_{\gamma }$ are recorded in Appendix C.6.

5.7. Verification of the Exponent

When the lower limit in the dissipation-rate condition is taken to $k_f\to 0$, the implicit condition simplifies and yields the closed-form expression

$$\beta ={\left[{\displaystyle \frac{2C_K\hspace{0.17em}\mathsf{\Gamma }(4/3m)}{m}}\right]}^{3m/4}.$$

(120)

Setting $C_K=1.5$ gives representative values: for $m=4/3$, $\beta =2.25$; for $m=3/2$, $\beta =2.37$. The corresponding energy spectra are

$$m={\displaystyle \frac{4}{3}}:E\left(k\right)=1.5\hspace{0.17em}{\epsilon }^{2/3}{k}^{-5/3}\exp \left[-2.25(k/k_d{)}^{{\displaystyle \frac{4}{3}}}\right],$$

(121)

$$m={\displaystyle \frac{3}{2}}:E\left(k\right)=1.5\hspace{0.17em}{\epsilon }^{2/3}{k}^{-5/3}\exp \left[-2.37(k/k_d{)}^{{\displaystyle \frac{3}{2}}}\right].$$

(122)

We emphasize the role of this comparison: within the present framework, the DNS evidence is not used to fit $m$ but to evaluate the plausibility of the admissible range (119) and to benchmark the resulting spectra against high-resolution data. The derivation of the $k_f\to 0$ simplification leading to (120) is given in Appendix C.7.

Ishihara et al. performed direct numerical simulations (DNS) of isotropic turbulence up to ${2048}^3$ grid points at $R{e}_{\lambda }=433$, and subsequently Kaneda and Ishihara extended the resolution to ${4096}^3$ at $R{e}_{\lambda }=675$ [37,38]. The energy spectra reported by Ishihara et al. exhibit a $-5/3$ power law in the inertial subrange (corresponding to the blue curve in Figure 1) and an exponential-type decay in the dissipation range.

In these DNS datasets, the effective resolution in $k/k_d$ reaches at most approximately $\sim 3$. Under this context, Figure 1 compares the DNS energy spectrum obtained with ${2048}^3$ grid points at $R{e}_{\lambda }=433$ against the spectrum predicted by the present model. In this study, we use $C_K=1.5$ as stated above. The proposed model (red curve in the figure) reproduces, with high accuracy, the spectral characteristics observed in DNS across both the inertial and dissipation ranges. For reference, we also plot the Pao model corresponding to $m=4/3$ within the present formulation; however, the Pao model tends to overestimate the energy spectrum in the dissipation range relative to the DNS results. These comparisons support the plausibility of the exponent range evaluated in this section. The same comparison has also been noted by Hebishima and Inage [39].

As evident from the figure, the energy spectra in the dissipation range corresponding to $m=4/3$ and $m=3/2$ exhibit only minor differences.

Next, we examine the sensitivity of the parameter $\beta$ to the Kolmogorov constant. In the present analysis, we adopt the commonly used value $C_K=1.5$. Experimental and numerical studies of high-Reynolds-number homogeneous isotropic turbulence, however, indicate that $C_K$ typically lies within the practical range $1.4\hspace{0.17em}-1.7$ [9,40]. To assess the robustness of the predicted parameter $\beta$ with respect to this uncertainty, we evaluate its variation over this range for two representative values of the tail exponent, $m=4/3$ and $m=3/2$.

The parameter $\beta$, defined in Equation (120), depends explicitly on the Kolmogorov constant $C_K$. Since $\beta \propto C_K^{3m/4}$, its variation with respect to $C_K$ is monotonic and follows a simple power-law scaling. Over the practical range $1.4\le C_K\le 1.7$, the resulting change in $\beta$ amounts to +21.43% for $m=4/3$ and +24.41% for $m=3/2$. Thus, although $\beta$ varies at the 20–25% level under plausible uncertainty in $C_K$, this dependence is smooth and does not indicate sensitivity to fine-tuned parameter choices.

Ishihara et al. aimed to clarify the structure of the energy spectrum in the near-dissipation range of homogeneous isotropic turbulence and to examine its Reynolds-number dependence using unprecedentedly high-resolution direct numerical simulations [41]. A series of DNS of the incompressible Navier–Stokes equations was performed with grid resolutions up to ${4096}^3$ and Taylor-scale Reynolds numbers up to $R_{\lambda }\approx 675$. This enabled a detailed statistical analysis of the dissipation-range spectrum through the quantity $D\left(k\right)=k^{2}E\left(k\right)$. By comparing spectra over a wide range of $R_{\lambda }$, the study showed that the dissipation-range peak height decreases monotonically with increasing Reynolds number, while the near-dissipation-range spectrum is well described by a stretched-exponential correction to inertial-range scaling. The parameters characterizing this correction vary with $R_{\lambda }$ but exhibit slow convergence toward asymptotic constants as $R_{\lambda }\to \mathrm{\infty }$. These results provide strong numerical evidence that, although inertial-range universality is approximately realized at high Reynolds numbers, the near-dissipation range retains a measurable and systematic Reynolds-number dependence.

Figure 2 compares $D\left(k\right)\equiv k^{2}E\left(k\right)$ obtained from the DNS of Ishihara et al. [41] with the prediction of the present model. The red solid curve represents the model result, while the colored dashed curves correspond to DNS at various Taylor-scale Reynolds numbers. In the DNS data, the dissipation-range peak height $I_p$, located near $k\eta \approx 0.2$, decreases monotonically as $R_{\lambda }$ increases. By contrast, the peak predicted by the present model appears slightly lower than those of the DNS cases shown. This discrepancy arises because the present model is constructed in the asymptotic limit $R_{\lambda }\to \mathrm{\infty }$. If DNS at substantially higher $R_{\lambda }$ become available, the difference in peak height is therefore expected to diminish. Regarding the peak location, the case $m=3/2$ shows better agreement with the DNS results.

On the low-wavenumber side ($k\eta <1$), the inertial subrange dominates. As $R_{\lambda }$ increases, the effective extent of the inertial subrange broadens, and the onset of $D\left(k\right)$ shifts toward smaller values of $k\eta$, a trend well established in DNS studies. Since the present model represents an asymptotic construction for $R_{\lambda }\to \mathrm{\infty }$, it reproduces this shift toward lower $k\eta$ in an asymptotic sense. When attention is restricted to the inertial subrange, the model therefore shows good agreement with DNS in both slope and overall spectral shape.

Recent high-resolution DNS provide strong physical support for the two-level relative-entropy theory developed in this study, including the cascade description based on a Fokker–Planck equation in logarithmic wavenumber space and the WKB-based selection mechanism for the dissipation-range spectral shape.

First, the fundamental assumption of cascade locality is strongly supported by the numerical analyses of Alexakis et al. [28] and Eyink and Aluie [26]. Using DNS at a resolution of ${1024}^3$, Alexakis et al. performed a shell-to-shell decomposition of the energy transfer and demonstrated that the dominant contribution to the total energy flux arises from interactions between adjacent scales, within the wavenumber ratio range $k^{\prime }/k\approx 1/2$ to 2. Direct energy exchange across widely separated scales therefore contributes negligibly to the statistical properties of the cascade. Complementarily, Eyink and Aluie established a rigorous form of multiscale locality through systematic coarse-graining analysis, showing that energy dissipation at a given scale is effectively decoupled from sufficiently larger or smaller scales. These results provide a solid dynamical foundation for coarse-graining Navier–Stokes triadic interactions into a local continuous jump process, represented here as a drift–diffusion process in logarithmic wavenumber space with an effective diffusion coefficient $D$.

Next, the dissipation-range spectral shape predicted by the present theory—the stretched-exponential form derived from WKB analysis and the Hyper-MaxEnt principle—shows close agreement with recent DNS results. Analyses by Verma [14] and Buaria et al. [12], based on DNS reaching $R_{\lambda }\approx 1300$, confirm that the dissipation-range spectrum is well described by $E\left(k\right)\propto \mathrm{e}\mathrm{x}\mathrm{p}[-\beta (k/k_d{)}^m]$. The exponent $m$ reported in these studies lies in the range $4/3\le m\le 3/2$, in contrast to the classical Pao model, which assumes a simple exponential form ($m=1$). The fact that the selection principle developed here yields an exponent $m^{\ast }$ within this DNS-supported range indicates that the present model captures the intrinsic coupling between information-theoretic optimality and viscous dissipation, rather than relying on empirical curve fitting. Physically, values $m>1$ reflect the competition between energy residence time and dissipation near the transition to the dissipation range.

The present theory also provides a natural interpretation of the bottleneck effect, a local spectral enhancement near the transition from the inertial to the dissipation range observed in DNS and experiments [42]. Within this framework, this non-universal feature arises from the dependence of spectral-shape selection on the prior distribution and boundary conditions entering the relative-entropy formulation. Reynolds-number dependence and flow-specific microstructures observed in DNS can thus be represented as variations in the statistical constraints of the theory.

Taken together, these results show that the proposed framework provides an integrated and physically grounded explanation of major DNS findings through two complementary principles: dynamical continuity, embodied in the Fokker–Planck evolution of the cascade, and statistical optimality, expressed through maximum-entropy selection. This synthesis establishes a coherent theoretical foundation for understanding the small-scale structure of turbulence.

5.8. Physical Validity of the Proposed Model and Its Application Across a Wide Range of Reynolds Numbers

Prominent anisotropic turbulence models, such as the RNG and TSDIA frameworks, are characterized by their systematic linkage of macroscopic anisotropy to the microscopic statistical symmetries of isotropic turbulence [43,44].

The RNG (Renormalization Group) model provides a statistical procedure for incorporating the nonlinear effects of small-scale eddies into an effective viscosity at larger scales. By iteratively eliminating high-wavenumber fluctuations, it reformulates energy transport as a correction to molecular viscosity and determines model constants through mathematical limiting operations rather than empirical tuning. Similarly, Yoshizawa’s TSDIA (Two-Scale Direct-Interaction Approximation) separates the macroscopic scale of the mean flow from the microscopic scale of turbulent fluctuations. Treating mean-flow strain and rotation as perturbations within the DIA framework, it derives anisotropic stress distributions, including the effects of streamline curvature, that are not captured by conventional eddy-viscosity models.

A common physical foundation of both frameworks is the assumption that Kolmogorov’s $-5/3$ law governs microscopic scales. Whether through the coarse -graining procedure in RNG or the expansion coefficients in TSDIA, the consistency of these models relies on the existence of a well-developed inertial subrange. This reflects the view that, even in strongly anisotropic flows, the small-scale energy-dissipating structures retain an approximately universal and isotropic character.

Models based on this universality, however, face an inherent limitation. They presuppose high-Reynolds-number flows in which the $-5/3$ law is realized over a sufficiently wide inertial range. At low-to-moderate Reynolds numbers, where this range is poorly developed, the theoretical basis of such models becomes less secure. Extending turbulence modeling across all Reynolds numbers therefore requires moving beyond inertial-range approximations and incorporating the viscous dissipation range. Describing the full spectral structure is not merely a mathematical refinement; it is essential for capturing the terminal mechanisms of the energy cascade that control predictive accuracy in nonequilibrium turbulent flows at lower Reynolds numbers.

From this perspective, Hebishima and Inage [39] evaluated an energy spectrum—validated using GOY shell-model results [45]—that bridges the inertial and dissipation ranges. By exploiting this extended spectral information, they proposed a model applicable over the entire Reynolds-number spectrum and showed a clear improvement in accuracy compared with conventional approaches.

The central contribution of the present framework is to provide a theoretical basis for such spectral information within an information-thermodynamic setting. Whereas previous functional forms describing the inertial-to-dissipation transition were largely empirical or heuristic, the present work establishes their validity through an H-theorem, a fundamental result of nonequilibrium statistical mechanics. By grounding the spectral transition in entropy production, this approach offers a non-empirical and physically robust foundation for turbulence modeling, with applicability spanning from the onset of turbulence to the high-Reynolds-number limit.

5.9. Summary: Connecting the First and Second Stages

In the first stage (dynamical level), within an autonomous conservative FP semigroup, the relative turbulence entropy is a Lyapunov functional and (90) holds. In the second stage (selection level), the WKB analysis constrains the dissipation tail to the stretched-exponential form (95) together with the coupling (102). Accordingly, we construct the global reference distribution $P_s^{\left(\gamma \right)}$ and define the final one-dimensional optimization

$${\gamma }^{\ast }\in \mathrm{a}\mathrm{r}\mathrm{g} \underset{\gamma \in \mathsf{\Gamma }}{\mathrm{m}\mathrm{i}\mathrm{n}}{\int }_{{\xi }_{-}}^{\mathrm{\infty }}{P}_{\gamma }\left(\xi \right)\mathrm{l}\mathrm{n}{\displaystyle \frac{P_{\gamma }\left(\xi \right)}{P_s^{\left(\gamma \right)}\left(\xi \right)}}\hspace{0.17em}d\xi ,$$

(123)

which identifies

$$m^{\ast }=1+{\displaystyle \frac{{\gamma }^{\ast }}{2}}.$$

(124)

Thus, the exponent $m$ is not an a posteriori fitting parameter but is positioned on a verifiable logical chain:

$$\begin{gathered} \left(\mathrm{irreversibility}: \mathit{H}-\mathrm{theorem}\right)\Rightarrow \left(\mathrm{dissipation} \mathrm{tail}: \mathrm{WKB}\right) \\ \Rightarrow \left(\mathrm{global} \mathrm{domain} \mathsf{\Omega }\right)\Rightarrow \left(1\mathrm{D} \mathrm{optimization} \mathrm{in} \gamma \right). \end{gathered}$$

6. Conclusions

This study proposed and validated a two-level entropy framework for isotropic turbulence spectra, in which dynamical admissibility and spectral selection are treated as logically distinct stages. This separation resolves a long-standing ambiguity in dissipation-range modeling by placing the stretched-exponential tail on a verifiable and non-empirical foundation.

At the dynamical level, a coarse-grained Fokker–Planck description in logarithmic wavenumber space was derived from the Lin equation. Under minimal and explicit assumptions, the relative entropy was shown to be a Lyapunov functional, thereby fixing an irreversible arrow of time in scale space. This result provides a rigorous dynamical constraint but, by construction, does not determine the stationary spectral shape of a nonequilibrium cascade.

At the selection level, the dissipation range was formulated as a stationary open-system problem with absorption. A WKB analysis constrained the admissible tail to stretched-exponential forms and established a direct coupling between the tail exponent and the high-wavenumber scaling of the effective diffusion. By enforcing dissipation-rate consistency and minimizing a one-dimensional Kullback–Leibler functional against a globally defined reference distribution, the remaining arbitrariness was removed. The dissipation-range exponent therefore emerges as a necessary consequence rather than a fitting parameter, and is found to agree quantitatively with high-resolution DNS.

From a conceptual standpoint, the present framework clarifies the role of entropy-based principles in turbulence theory. Entropy does not replace dynamics; instead, it selects among asymptotic states that are dynamically admissible. The stretched-exponential dissipation tail is thus interpreted as the statistically most stable realization compatible with irreversible cascade dynamics and asymptotic transport constraints.

The framework is deliberately minimal and extensible. Natural directions include extensions to anisotropic and shear-driven turbulence, scalar mixing, and time-dependent or inhomogeneous cascades, where the same separation between dynamical irreversibility and entropic selection is expected to remain operative. In this sense, the present theory provides not a closed model, but a structural principle for turbulence spectra beyond isotropy.