Generalized Hydrodynamic Kinetic Equations

• Generalized hydrodynamic kinetic equations are a systematic framework that couples kinetic distribution functions with high-order moment hierarchies to capture multiscale, far-from-equilibrium phenomena.
• The methodology integrates spectral, perturbative, and machine learning approaches to provide closure-free, physically-parametrized models that link microscopic dynamics to macroscopic observables.
• These equations account for nonlocal effects, nonlinearities, and fluctuating hydrodynamics, offering improved predictions in classical, quantum, and active matter systems.

Generalized hydrodynamic kinetic equations constitute a broad, technically rigorous framework for describing nonequilibrium transport and collective phenomena in classical, quantum, and active matter, integrating and extending traditional hydrodynamics and kinetic theory. They arise from systematically coupling the evolution of phase-space distribution functions (kinetic description) to multi-scale, high-order moments and are structured to capture nonlocal effects, nonlinearities, fluctuating hydrodynamics, and situations far from local equilibrium. This article surveys the foundational advances, mathematical structures, and key implications across kinetic, hydrodynamic, and statistical frameworks as reflected in landmark research.

1. Hierarchical Structure and Systematic Generalization

A defining feature of generalized hydrodynamic kinetic equations is their construction as infinite or high-order hierarchies for distribution moments (mass, momentum, energy, fluxes of all orders), departing from the finite closure of conventional hydrodynamics (e.g., Navier-Stokes–Fourier equations). In the Nonequilibrium Statistical Ensemble Formalism (NESEF), the evolution of the single-particle distribution function $f_1(\mathbf{r},\mathbf{p},t)$ leads systematically to the coupled moment hierarchy:

$$I_n^{[r]}(\mathbf{r}, t) = \int d^3p\, u_n^{[r]}(\mathbf{p}) f_1(\mathbf{r},\mathbf{p},t)$$

with

$$\frac{\partial}{\partial t} I_n^{[l]} + \nabla \cdot I_n^{[l+1]} = \mathcal{S}_n^{[l]}[I, f_1, ...]$$

where $\mathcal{S}_n^{[l]}$ contains collision, relaxation, and nonlinear coupling terms, often highly nonlocal and noninstantaneous (Silva et al., 2012). This closure-free description naturally accounts for strong inhomogeneity, steep gradients, and processes with large Knudsen numbers.

Crucially, the moment equations can describe both macroscopic (hydrodynamic) and microscopic (kinetic) regimes, and the contraction of description—i.e., deciding up to which moment to truncate—becomes a function of characteristic timescales:

$$\theta_l^{-1} = l\left[|a_{\tau 0}| + (l-1)|b_{\tau 1}|\right]$$

These "Maxwell-times" form a hierarchy paralleling the Bogoliubov picture and rationalize systematic model reduction (Silva et al., 2012, Rodrigues et al., 2019).

2. Hybrid and Physically Parametrized Model Equations

Explicit models such as the Holway–Shakhov equation provide a bridge between kinetic and hydrodynamic regimes by uniting the strengths of various relaxation-time approximations. This hybrid equation,

$$\mathbf{C}\frac{\partial h}{\partial \mathbf{r}} = \nu\sqrt{\beta} [\dots + \gamma \mathbf{C}(C^2 - \tfrac{5}{2})\mathbf{Q} + \omega (C_i C_j - \tfrac{1}{3}\delta_{ij}C^2) P_{ij}],$$

includes parameters $(\nu, \omega, \gamma)$ that are uniquely and directly determined by measured transport coefficients: diffusion $D$, viscosity $\eta$, and thermal conductivity $\varkappa$, as well as dimensionless numbers such as Prandtl and Ferziger numbers (Latyshev et al., 2013). This explicit parametrization, completed via collision integral brackets $\Omega^{(i,j)}$ reflecting the intermolecular potential, enables physically realistic modeling of rarefied gas phenomena and analytical boundary-layer solutions:

Parameter	Relation to Transport Coefficients
$\nu$	$\nu = \frac{kT}{mD}$
$\omega$	$\omega = 2(1 - D/\nu_*)$
$\gamma$	$\gamma = \frac{4}{5}(1 - \mathrm{Pr}\,\mathrm{Fe})$

This model ground-anchors generalized hydrodynamics in experimental observables, mitigating the arbitrariness of previous kinetic parameterizations.

3. Nonlocal, Nonlinear, and Dissipative Generalizations

The scope of generalized hydrodynamic kinetic equations extends to systems with non-Markovian memory, strong correlations, and far-from-equilibrium fluctuations. Formulations derived via the Zubarev nonequilibrium statistical operator (NSO) framework yield kinetic equations with explicit time-nonlocal (memory) integrals and coupled evolution for both kinetic and collective (hydrodynamic) variables (Markiv et al., 2013, Hlushak et al., 2015). For quantum Bose systems and classical liquids, this structure naturally bridges the BBGKY hierarchy, kinetic-level distribution functions, and generalized Fokker-Planck equations for hydrodynamic variables, incorporating nonlinear drift and diffusion governed by time-correlated fluxes:

$$\frac{\partial f(a; t)}{\partial t} = -\sum_{l,\mathbf{k}} \frac{\partial}{\partial a_{l,\mathbf{k}}}[v_{l,\mathbf{k}}(a;t) f(a;t)] + \cdots$$

where both $v_{l,\mathbf{k}}$ and the generalized diffusion coefficients are functionals of the collective variable distributions and their high-order cumulants (Hlushak et al., 2015, Hlushak et al., 2013).

In low-dimensional quantum integrable systems, kinetic equations for the quasiparticle distribution (filling function) are strongly nonlocal and nonlinear, featuring "dressing" transformations and non-local diffusion or collision integrals that encode the infinite set of conservation laws and break integrability (Møller et al., 2022):

$$\partial_t f + v^{\mathrm{eff}}[f]\, \partial_z f + a^{\mathrm{eff}}[f]\, \partial_\theta f = \mathcal{N}[f]$$

with $\mathcal{N}[f]$ representing either quantum Boltzmann-type collision or nonlinear, nonlocal diffusion (Møller et al., 2022).

4. Methodologies: Spectral, Perturbative, and Non-Perturbative Approaches

The derivation and analysis of generalized hydrodynamic kinetic equations leverage several complementary techniques:

• Spectral theory and splitting: Unified spectral approaches elucidate why diverse kinetic models (Boltzmann, Landau, quantum) universally reduce to the Navier-Stokes-Fourier system as $\varepsilon \rightarrow 0$, under minimal structural assumptions (mass, momentum, and energy conservation; spectral gap) (Gervais et al., 2023). The spectral decomposition segregates slow hydrodynamic modes (diffusive, acoustic) from fast-decaying kinetic modes, enabling robust convergence and quantification in strong topologies.
• Perturbative expansions: Chapman-Enskog, Hilbert, and modern BDNK moment-hierarchy expansions systematically approximate kinetic equations in small Knudsen number, with care taken for matching conditions (hydrodynamic frames), which critically impact causality and stability in relativistic settings (Rocha et al., 2022). For relativistic hydrodynamics, only certain frame choices (e.g., BDNK) ensure stable, causal evolution at first order.
• Nonperturbative invariance and closure: Dynamic invariance equations, structurally analogous to Schwinger-Dyson equations, enable non-perturbative, closed-form expressions for hydrodynamic manifolds beyond the limitations of gradient expansions (Chapman-Enskog). The so-called "pullout" procedure delivers analytic, bounded, and globally valid closures for hydrodynamic frequencies, overcoming the instability and unphysical divergence of traditional Burnett or super-Burnett expansions (Karlin et al., 2013).

5. Stochastic and Fluctuating Hydrodynamics

Generalized hydrodynamic kinetic approaches encode the influence of mesoscopic and thermal fluctuations on macroscopic evolution. Hydro-kinetic equations for the correlators (two-point functions) of conserved variables capture the long-time tails, renormalization of transport coefficients, and the emergence of nonanalytic, fractional power temporal scaling in the energy-momentum tensor, especially under expansion (Bjorken flow) or near critical points (Akamatsu et al., 2016, Akamatsu et al., 2017).

The hierarchy for fluctuation correlators $N_{AA}(\tau, \mathbf{k})$ admits:

$$\partial_\tau N_{AA} = -\Gamma_A(\tau,k) [N_{AA} - N_0(\tau)] - \text{driving terms}$$

yielding corrections to observables (e.g., longitudinal pressure) that scale as inverse fractional powers of time, e.g., $\sim \tau^{-3/2}$, and are quantitatively dominant over second-order viscous hydrodynamic corrections in heavy-ion collisions. This formalism enables the modular coupling of hydro-kinetic equations with existing hydrodynamic simulation codes for improved realism (Akamatsu et al., 2016, Akamatsu et al., 2017).

6. Machine-Learned and Data-Driven Closures

Recent advances utilize machine learning to construct uniformly accurate reduced-order models for generalized hydrodynamic equations, overcoming the closure problem for moment systems without assuming scale separation (Han et al., 2019). By learning optimal generalized moments (including Galilean-invariant moments) and closure relations through autoencoders and neural networks, data-driven frameworks replace ad hoc or analytically intractable closures. Active learning algorithms guide the sampling of kinetic state space, ensuring coverage and robustness across all Knudsen number regimes. Such models yield discretization-independent, physically constrained (symmetry-preserving), and interpretable hydrodynamic PDE systems with uniform accuracy across the kinetic-hydrodynamic transition.

Step	Implementation
Moment selection	Neural network encoder
Closure construction	ML-learned fluxes/relaxation terms
Physical constraints	Explicit via invariance or loss functions
Data exploration	Error-driven active sampling
Numerical efficiency	Simulation speed-up over kinetic equation

7. Extensions: Quantum, Nonlinear Fluctuations, and Integrable Systems

Quantum kinetic theory generalized from quantum hydrodynamics employs operator-based distribution functions, leading to coupled kinetic equations for charge, spin, dipole, and multi-component (e.g., sublattice, Dirac) degrees of freedom; quantum correlations naturally arise via two-particle distribution functions (Andreev, 2012). Nonlinear hydrodynamic fluctuations, organization of the hierarchy via cumulant (beyond Gaussian) expansions, and consistent handling of multi-particle, non-Gaussian distributions with generalized Fokker-Planck equations, support the rigorous description of quantum turbulence, phase transitions, and anomalous transport (Hlushak et al., 2013, Hlushak et al., 2015).

In classical and quantum integrable systems, the generalized hydrodynamic structure accommodates nonlocal, nonlinear advection and dissipation terms, captures the role of the infinite set of conservation laws, and admits precise high-order numerical schemes based on backward semi-Lagrangian and time-Taylor expansions, necessary for simulating ballistic and weakly non-integrable dynamics (Møller et al., 2022).

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These unified developments in generalized hydrodynamic kinetic theory establish rigorous, physically parametrized, and often computationally tractable frameworks. They are essential for the quantitative description and prediction of far-from-equilibrium dynamics, nonlocal transport, collective behavior in both classical and quantum systems, and for addressing the multiscale, fluctuating, and strongly interacting nature of modern soft and active matter, plasmas, and complex fluids.