Geometric Decomposition of Entropy Production

• Geometric decomposition of entropy production is a method that partitions entropy functionals in nonequilibrium kinetic models, revealing both microscopic and mesoscopic dissipation.
• It employs geometric and spectral techniques to separate contributions based on convexity and tensor analysis, ensuring all dissipation terms remain nonnegative.
• The approach underpins the H-theorem and guarantees functional compactness, which is critical for proving the existence of weak solutions in polyatomic gas dynamics.

Geometric decomposition of entropy production refers to the rigorous partitioning of entropy production functionals—quantifying irreversibility in nonequilibrium kinetic equations or stochastic processes—into terms that are structurally and mathematically distinct, often with origins in convexity, information geometry, or spectral analysis. In kinetic theory, this approach provides sharper understanding and control of dissipation, the $H$-theorem, and compactness properties in models beyond the classical BGK scenario.

1. Polyatomic ES-BGK Kinetic Model and Entropy Production

The polyatomic ellipsoidal Bhatnagar–Gross–Krook (ES-BGK) model describes the relaxation of a molecular distribution function $f = f(x, v, I, t)$, where $v \in \mathbb{R}^3$ is velocity and $I$ denotes extra internal degrees of freedom, such as rotation and vibration, present in polyatomic gases. The model takes the form: $$\partial_t f + v \cdot \nabla_x f = \mathcal{A}_{v, \theta} \left( \mathcal{M}_{v, \theta}(f) - f \right)$$ where $\mathcal{M}_{v, \theta}(f)$ is a local polyatomic Maxwellian, encoding selected macroscopic moments (density, momentum, energy, and higher-order internal modes).

The associated entropy production functional is: $$D_{v, \theta}(f) = -\int_{\mathbb{R}^3 \times \mathbb{R}_+} \left( \mathcal{M}_{v,\theta}(f) - f \right) \ln f \; dv dI$$ The sign and structure of $D_{v, \theta}(f)$ are crucial for thermodynamic consistency—the $H$-theorem and weak compactness of solution sequences.

2. Positive Decomposition: Mathematical Structure

A key result for the polyatomic ES-BGK model is the discovery that $D_{v, \theta}(f)$ can be expressed as the sum of two nonnegative functionals: $$D_{v,\theta}(f) = \int \left( \mathcal{M}_{v,\theta}(f) - f \right) \left( \ln \mathcal{M}_{v,\theta}(f) - \ln f \right) dv dI + R_{v,\theta}$$ with

$$R_{v,\theta} = \int (\mathcal{M}_{v,\theta}(f) - f) \left[ \frac{(v-U)^T \mathcal{T}_{v,\theta}^{-1}(v-U)}{2} + \frac{I^{2/\delta}}{T_\theta} \right] dv dI$$

where $U$ is the local mean velocity, $\mathcal{T}_{v,\theta}$ is the effective temperature tensor, and $T_\theta$ is a temperature parameter associated with internal energy degrees of freedom. The main theorem asserts that for $0 < v < 1$, $0 < \theta < 1$ (the physically relevant range), both terms are nonnegative.

The non-negativity of the first term follows immediately from the strict convexity of $x \ln x$, as in the classical $L^1$-contraction and entropy dissipation analysis of the original BGK model. The remainder $R_{v,\theta}$ is nontrivial—its sign relies on detailed spectral decomposition of the temperature tensor and moment coefficients, and on convexity arguments specific to polyatomic gas dynamics and the structure of $\mathcal{M}_{v, \theta}$. This is rigorously established in Lemmas 2.1–2.3 and the main theorem.

3. Applications: H-Theorem and Weak Compactness

The decomposition enables a direct proof of the $H$-theorem for the polyatomic ES-BGK model: $$\frac{d}{dt} H(f) \leq 0, \qquad H(f) := \int f \ln f \, dv dI$$ since all contributions to $D_{v,\theta}(f)$ are manifestly nonnegative. Physical entropy is thus strictly nonincreasing along trajectories (with equality only at equilibrium), matching fundamental expectations for irreversible kinetic systems.

Beyond this, the decomposition is critical for functional analytic compactness. The $L^1$-weak compactness of the relaxation operator

$f \mapsto \mathcal{M}_{v, \theta}(f)$

depends on the control of moment growth. The entropy decomposition bounds higher moments (via $R_{v,\theta}$), which ensures that the integral operator cannot diverge; this is essential for establishing existence of weak solutions and for passing to strong limits in approximation sequences, leveraging Dunford–Pettis compactness.

4. Geometric and Spectral Interpretation

The positive decomposition has a geometric underpinning:

• The first term measures the "distance" (in relative entropy) from the actual distribution $f$ to the local polyatomic Maxwellian, capturing the dissipation due to local non-Maxwellian structure.
• The term $R_{v,\theta}$ measures the deviation of macroscopic moments (second moments of $v$ and internal energies) from their target equilibrium values—the lack of alignment of $f$’s macroscopic tensorial quantities with those prescribed by $\mathcal{M}_{v, \theta}(f)$.

This reflects a two-level geometric structure: $f$ is not only distinguished from $\mathcal{M}_{v, \theta}$ in the $L^1$-relative entropy "distance," but also in the space of macroscopic moment tensors. The spectral decomposition of the stress/temperature tensor enters critically in proving positivity of $R_{v,\theta}$, emphasizing the inherently geometric (in tensor space) meaning of entropy production for polyatomic kinetic systems.

5. Summary Table: Decomposition Structure

Term	Formula	Non-negativity	Geometric Role
Total entropy prod.	$D_{v,\theta}(f)$	Yes	Decay of total system entropy
Main dissipation	$$\int [\mathcal{M}_{v,\theta}(f) - f][\ln \mathcal{M}_{v,\theta}(f) - \ln f] dv dI$$	Yes	Distance from $f$ to Maxwellian
Remainder	$R_{v,\theta}$ (see above)	Yes	Distance of moments to equilibrium

6. Broader Context and Implications

This positive geometric decomposition is a technical advance over the monatomic BGK case: the polyatomic and ES features introduce nontrivial tensorial corrections, so simple entropy production calculations no longer immediately imply the H-theorem or compactness. The two-term structure reveals a hierarchy of irreversibility mechanisms: microscopic (distributional) and mesoscopic (moment-tensor) deviation from equilibrium.

This sharp functional control is not only vital for kinetic theory but informs nonequilibrium statistical mechanics more broadly, as similar geometric decompositions underlie entropy production analyses in Markov processes, information geometry of thermodynamic flows, and structure-preserving models with tensor-valued degrees of freedom. The extension to polyatomic gases suggests general pathways for rigorous entropy production analysis in complex kinetic models.

7. Conclusion

The geometric decomposition of entropy production in the polyatomic ES-BGK model rigorously partitions the entropy dissipation into nonnegative, physically and mathematically distinct contributions associated with distributional and tensorial distances to equilibrium. This decomposition underpins the mathematical proof of the H-theorem, enables compactness methods required for existence analysis, and clarifies the deeper geometric structure of irreversibility in advanced kinetic models for polyatomic gases. The methodology invites further generalizations both within kinetic theory and in structurally analogous nonequilibrium systems.