14-Moment Approximation in Kinetic Theory

• The 14-moment approximation is a moment-closure method that captures non-equilibrium effects by deriving a finite set of hyperbolic PDEs from the first 14 velocity moments.
• It employs a maximum-entropy principle to generate an exponential-polynomial velocity distribution, ensuring physical realizability and improved accuracy over lower-order models.
• This framework is instrumental in modeling rarefied flows, magnetized plasmas, and relativistic hydrodynamics, offering enhanced stability and predictive capability in high-Knudsen regimes.

The 14-moment approximation is a systematic moment-closure framework for kinetic equations, yielding a finite set of hyperbolic PDEs for non-equilibrium gas and plasma systems. It constructs a closed dynamical description based on the first fourteen velocity moments—including mass, momentum, stress, heat flux, and a contracted fourth-order moment—of the single-particle distribution function. The closure is achieved most rigorously by maximizing a kinetic entropy under these moment constraints, leading to a velocity distribution of exponential-polynomial form. This methodology, originally developed in classical and relativistic contexts, underpins both the Israel–Stewart theory of dissipative relativistic hydrodynamics and the maximum-entropy fluid models for rarefied or magnetized plasmas and gases. The 14-moment system bridges the gap between classical hydrodynamics and the full kinetic theory, capturing important non-equilibrium phenomena that lower-order closures (such as Euler or Navier–Stokes) cannot represent.

1. Definition and Structure of the 14 Moments

The 14-moment method is built upon the evaluation of the first fourteen moments of the velocity distribution function $f(\mathbf v)$. In the non-relativistic (Boltzmann) case for three dimensions, these are:

• Mass density: $\rho = \int m f(\mathbf v)\, d^3v$
• Momentum density: $\rho u_i = \int m v_i f(\mathbf v)\, d^3v$
• Pressure tensor: $P_{ij} = \int m c_i c_j f(\mathbf v)\, d^3v$ with $c_i = v_i-u_i$
• Heat flux vector: $q_i = \int m c_i c^2 f(\mathbf v)\, d^3v$
• Contracted 4th moment (kurtosis): $$R_{i i j j} = \int m c_i^2 c_j^2 f(\mathbf v)\, d^3v$$

Counting degrees of freedom (for 3D): 1 (mass) + 3 (momentum) + 6 (symmetric $P_{ij}$) + 3 (heat flux) + 1 (scalar fourth moment), generating a 14-dimensional dynamical field (Boccelli et al., 2024, Boccelli et al., 2024, Boccelli et al., 2021, Boccelli et al., 2023, Boccelli et al., 2022).

In the relativistic context (Grad's approach, Israel–Stewart formalism), an analogous 14 independent moments are constructed from the irreducible projections of the energy-momentum tensor and the particle (charge) current: number and energy densities, fluid velocity components, bulk viscous pressure, heat flux (or diffusion current), and shear-stress tensor components (Denicol et al., 2012, Denicol et al., 2018, Denicol et al., 2014, Takamoto et al., 2010).

2. Maximum-Entropy Closure and Distribution Ansatz

The central tenet of the 14-moment maximum-entropy approach is to determine, among all physically admissible velocity distribution functions $f(\mathbf v)$ reproducing a given set of 14 moments, the unique function that maximizes the kinetic Boltzmann entropy: $$S[f] = -k_B \int f \ln\left(\frac{f}{y}\right) d^3v$$ subject to

$$\int \Phi_a(\mathbf v) f(\mathbf v) d^3v = U_a\quad(a=1,\dots,14)$$

with basis functions $\Phi = \{1,\, v_i,\, v_i v_j,\, v_i v^2,\, v^4\}$.

The maximizer is an exponential of a polynomial in velocities up to fourth order: $$f_\mathrm{ME}(\mathbf v) = \exp\big(\lambda_0 + \lambda_i v_i + \lambda_{ij} v_i v_j + \lambda_{i,3} v_i v^2 + \lambda_4 v^4\big)$$ with 14 Lagrange multipliers determined implicitly by the nonlinear moment constraints above (Boccelli et al., 2024, Boccelli et al., 2021, Boccelli et al., 2024).

This closure ensures $f_\mathrm{ME} \geq 0$ and produces a strictly hyperbolic PDE system, in contrast to polynomial-based closures (e.g., original Grad 13-moment, which may yield negative distributions and parabolicity) (Boccelli et al., 2023, Boccelli et al., 2022).

3. Governing Equations and Moment Hierarchy

Applying the 14-moment closure to the kinetic equation (e.g., the Vlasov–Boltzmann equation, or its relativistic generalizations) leads to a set of coupled hyperbolic PDEs: $$\partial_t U_k + \nabla \cdot F_k(\mathbf U) = S_\mathrm{em, k}(\mathbf U) + S_{c, k}(\mathbf U)$$ where $U_k$ represent the 14 retained moments, $F_k$ their fluxes, and $S$ source terms (e.g., electromagnetic, collisional, or in ion models, ionization) (Boccelli et al., 2021, Boccelli et al., 2022).

The structure of the fluxes involves higher-order moments:

• $Q_{ijk} = \int m c_i c_j c_k f(\mathbf v) d^3v$
• $R_{ijkk}$, $S_{ijjkk}$ and others

These are not retained as independent fields but are calculated as algebraic functions of the 14 primary moments by direct integration of the maximum-entropy ansatz (or, in practice, using interpolative closure formulas to avoid high-dimensional integration at runtime) (Boccelli et al., 2024, Boccelli et al., 2023).

The maximal hyperbolicity of the closed system is a major mathematical advantage, enabling robust numerical shock-capturing and ensuring finite signal propagation speed, which is crucial for physical fidelity in non-equilibrium regimes (Boccelli et al., 2024, Boccelli et al., 2021, Takamoto et al., 2010).

4. Applications, Transport Coefficients, and Physical Regimes

Classical gases/plasmas: The 14-moment system, in its BGK-collision or Vlasov limit, is widely used to capture rarefied, non-equilibrium, or strongly magnetized flows where Fourier heat flux and Newtonian stress laws break down. In Hall thruster electron modeling, for example, the method captures ring-shaped velocity distributions, strong anisotropies, and non-equilibrium heat fluxes out of reach for lower-order closures; accuracy is demonstrated to within 5–10% of full kinetic or particle-in-cell solutions in relevant test cases (Boccelli et al., 2021, Boccelli et al., 2022).

High-Knudsen number rarefied flows and supersonic crossflows: The 14-moment closure reproduces both shock and wake structures and the underlying VDFs more accurately than 5- or 10-moment models, especially at Knudsen numbers $\geq 0.1$, and maintains hyperbolicity and realizability up to the so-called Junk manifold (Boccelli et al., 2023, Boccelli et al., 2024).

Relativistic hydrodynamics: In the Israel–Stewart theory and its extensions, the 14-moment approximation serves as the canonical closure for dissipative hydrodynamics, introducing relaxation-type (second-order) PDEs for bulk pressure, shear stress, and heat flux/diffusion, with well-defined transport coefficients (e.g., viscosities, conductivities) computable from the closure (Denicol et al., 2012, Takamoto et al., 2010, Denicol et al., 2014, Denicol et al., 2018).

Transport coefficients, such as shear and bulk viscosities and their associated relaxation times, can be explicitly expressed within the 14-moment framework in terms of equilibrium thermodynamic integrals. Notably, the temperature dependence and coupling terms (e.g., bulk–shear coupling) display nontrivial scaling and can dominate the evolution of certain dissipative channels (Denicol et al., 2014).

Magnetohydrodynamics: In non-resistive, dissipative MHD, the 14-moment system enables a causal, hyperbolic treatment, properly encoding anisotropic transport due to magnetic fields, and is compatible with both first- and second-order (Navier–Stokes and Israel–Stewart) limits (Denicol et al., 2018, Boccelli et al., 2021).

5. Numerical Implementation, Hyperbolicity, and Limitations

The 14-moment PDEs are discretized using finite-volume, Godunov-type schemes, and explicit treatment of source/relaxation terms is typical. Approximate closures for higher moments have been developed (e.g., McDonald–Torrilhon interpolants), allowing for efficient evaluation without high-dimensional nonlinear root-solving (Boccelli et al., 2024, Boccelli et al., 2022, Boccelli et al., 2023).

GPU implementation further accelerates computation, with speed-ups on the order of 20–340× (depending on hardware), making high-dimensional or multi-species 14-moment calculations tractable for applications in flow or plasma simulations (Boccelli et al., 2024).

The 14-moment approximation is mathematically guaranteed to be globally hyperbolic and strictly realizable (positive-definite VDF) as long as the moment set does not approach the Junk manifold—a singular limit at the boundary of the realizable moment-space, beyond which no positive entropy-maximizing VDF exists. In practical computation, proximity to this subspace is monitored and limited by adaptive dissipation or regularization (Boccelli et al., 2024, Boccelli et al., 2022, Boccelli et al., 2023).

Limitations: Although the 14-moment model captures key non-equilibrium effects absent in lower-order closures, it cannot reproduce all spectral features of the full kinetic theory, particularly for high-frequency, short-wavelength perturbations (continuous kinetic branches, damping singularities); these require higher-moment or alternative nonlocal closures. The 14-moment system is, however, a substantial improvement over Navier–Stokes in causality and stability (Takamoto et al., 2010, Denicol et al., 2012, Jaiswal, 2013).

6. Comparison with Alternative Closures and Physical Realizability

The 14-moment closure, especially in its maximum-entropy form, overcomes several deficiencies of the original Grad expansion and related polynomial closures. Key points of comparison include:

• Grad 13-moment polynomial ansatz: Not guaranteed to yield positive distributions, leading to unphysical negative densities and the breakdown of hyperbolicity/parabolicity. In contrast, the exponential-polynomial (maximum-entropy) form always produces a nonnegative VDF (Boccelli et al., 2024, Boccelli et al., 2022, Boccelli et al., 2023).
• Hierarchical extensibility: While 14 moments are natural for systems with stress and heat-flux but no internal energy exchange or detailed chemistry, the framework generalizes to higher-order (e.g., 21-moment) maximum-entropy closures to capture more detailed features such as multiple beams, more pronounced tails, or complex anisotropies (Boccelli et al., 2024).
• Realizability and the Junk manifolds: The mathematical domain in which a positive VDF exists is nontrivial but is completely characterized for the 14-moment set, with singular limits corresponding to non-Maxwellian distributions such as collapsing shells and lobe/beam structures (Boccelli et al., 2024, Boccelli et al., 2022).

These properties render the 14-moment maximum-entropy closure a foundational tool in non-equilibrium gas and plasma modeling, facilitating accurate, hyperbolic, and realizable dynamics across a remarkably broad parameter regime.

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• (Boccelli et al., 2021) A 14-moment maximum-entropy description of electrons in crossed electric and magnetic fields
• (Boccelli et al., 2024) A gallery of maximum-entropy distributions: 14 and 21 moments
• (Boccelli et al., 2024) Numerical simulation of rarefied supersonic flows using a fourth-order maximum-entropy moment method with interpolative closure
• (Boccelli et al., 2022) 14-moment maximum-entropy modelling of collisionless ions for Hall thruster discharges
• (Boccelli et al., 2023) Modelling high-Mach-number rarefied crossflows past a flat plate using the maximum-entropy moment method
• (Denicol et al., 2014) Transport Coefficients of Bulk Viscous Pressure in the 14-moment approximation
• (Denicol et al., 2012) Derivation of fluid dynamics from kinetic theory with the 14--moment approximation
• (Jaiswal, 2013) Relativistic dissipative hydrodynamics from kinetic theory with relaxation time approximation
• (Denicol et al., 2018) Non-resistive dissipative magnetohydrodynamics from the Boltzmann equation in the 14-moment approximation
• (Takamoto et al., 2010) The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar-Gross-Krook model and Grad's 14-moment expansion