Hydrodynamic-Frame-Robust Regime

• Hydrodynamic-Frame-Robust Regime is a framework ensuring macroscopic consistency despite changes in local flow, temperature, and chemical potential definitions.
• It applies to diverse systems like Fermi gases, electron fluids, and anisotropic liquids, providing robust constitutive relations across varying collisionalities and boundary conditions.
• The regime is validated through gradient expansion and moment methods, effectively capturing crossovers from collisionless to hydrodynamic behaviors with clear spectral signatures.

The hydrodynamic-frame-robust regime characterizes the class of physical and mathematical descriptions of continuum systems—ranging from quantum Fermi gases and electron fluids to complex anisotropic liquids—where the constitutive relations, collective modes, and observables exhibit invariance or controlled dependence under changes of the hydrodynamic frame: the operational definitions of local flow velocity, temperature, and chemical potential. This regime is distinguished by the ability of hydrodynamic, kinetic, and generalized constitutive frameworks to consistently capture macroscopic dynamics and spectral features across a range of collisionalities, interaction strengths, and geometrical constraints, and to remain robust under changes to the representation of non-equilibrium degrees of freedom or internal rotational symmetries.

1. Definition and Physical Context

In hydrodynamic modeling, the “frame” refers to choices in defining the fluid’s local four-velocity and associated thermodynamic variables out of equilibrium. Physical observables, such as the dynamic structure factor or transport coefficients, should be insensitive to the ambiguity of this choice provided the underlying effective description is robust. The hydrodynamic-frame-robust regime arises when the system is sufficiently close to local equilibrium—typically ensured by rapid local interactions, such as electron-electron or magnon-magnon collisions—so that the macroscopic equations admit a consistent, objective description under allowable field redefinitions or symmetry operations.

Typical systems manifesting such robust regimes include

• Normal Fermi gases traversing from collisionless to hydrodynamic domains (Watabe et al., 2010, Narushima et al., 2017);
• Graphene and 2D Fermi liquids under electronic hydrodynamics, incorporating boundary effects (Wagner, 2015, Chandra et al., 2018);
• Chiral and topological systems (e.g., Weyl semimetals) where hydrodynamic and anomalous/nonlocal responses coexist (Gorbar et al., 2018, Messica et al., 2022);
• Biaxial nematic liquid crystals modeled with full SO(3) frame dynamics (Li et al., 2022, Feng et al., 8 Aug 2025);
• Magnon fluids exhibiting collective transport (Sano et al., 2022);
• Relativistic viscous fluids and quark–gluon plasmas under varying choices of hydrodynamic variables or generalized frames (Noronha et al., 2021, Diles et al., 2023, Bhattacharyya et al., 2023, Bhambure et al., 13 Dec 2024).

2. Mathematical Structures and Constitutive Frameworks

Mathematically, the hydrodynamic-frame-robust regime is realized in models whose constitutive relations can be transformed under a general set of gradient-dependent variable redefinitions without altering the physical content of the linearized hydrodynamic modes: $$u^{\mu\prime} = u^\mu + \delta u^\mu, \quad T' = T + \delta T, \quad \mu' = \mu + \delta\mu,$$ with $\delta u$, $\delta T$, $\delta\mu$ of order one or more derivatives. In relativistic hydrodynamics, “frame” changes correspond to shifting dissipative corrections among terms of the stress tensor and current, yet observables such as the shear and sound mode dispersion relations can be reparametrized in terms of frame-invariant combinations of transport coefficients (Diles et al., 2023).

The general approach is encapsulated in systematic gradient expansions to higher (e.g., third) order and by algorithms (e.g., the Irreducible-Structure (IS) algorithm) that build all allowed terms from derivatives of the basic fluid and symmetry fields. For systems with internal symmetry structures (e.g., the rotational SO(3) invariance of biaxial nematics), the inclusion of rotationally covariant derivatives is central to defining frame-robust models (Feng et al., 8 Aug 2025).

Kinetic approaches, particularly those based on moment methods for the Boltzmann equation, have demonstrated that a unified description can interpolate between collisionless (zero sound) and hydrodynamic (first sound) regimes, reconciling macroscopic hydrodynamic predictions with the dynamics dominated by mean-field or collective effects (Watabe et al., 2010, Narushima et al., 2017). In such cases, the spectral response encoded in dynamic structure factors $S(q, \omega)$ is shown to evolve smoothly independent of the specific implementation details of the hydrodynamic fields.

3. Crossover Regimes and Emergent Collective Dynamics

A hallmark of the hydrodynamic-frame-robust regime is the continuous crossover in collective excitation spectra and transport signatures as the system transitions from collisionless (ballistic or mean-field-dominated) to hydrodynamic (strongly collisional and locally equilibrated) behavior. This is evident in several contexts:

• Normal Fermi gases: The dynamic structure factor exhibits a crossover of the Brillouin peak from zero sound (collisionless) to first sound (hydrodynamic), with the emergence of a Rayleigh peak (thermal diffusion) at low frequency, and quantitative features controlled by temperature and interaction strength (Watabe et al., 2010).
• Imbalanced mixtures: Mass or population imbalance in two-component Fermi gases shifts collective mode frequencies, modifies the crossover temperature scale, and introduces unique spin responses in the dynamic structure factor, all captured in a hydrodynamic-frame-robust formalism via the moment method (Narushima et al., 2017).
• Magnon fluids: In hydrodynamic regimes, heat and spin conductivities become decoupled due to different momentum relaxation processes, resulting in a breakdown of the magnonic Wiedemann-Franz law as a definitive marker for hydrodynamic magnon transport (Sano et al., 2022).
• Hydrodynamics with slow modes: Near critical points, additional slow non-hydrodynamic modes regularize critical slowing down, ensuring a non-divergent bulk viscosity and hydrodynamic response. The effective theory (“Hydro+”) explicitly includes such modes, deriving relations where sound attenuations and transport coefficients remain finite and robust (Stephanov et al., 2017).

This crossover can often be experimentally probed, for example, in ultracold Fermi gases using Bragg spectroscopy or in electron fluids via AC transport and nonlocal resistance measurements (Watabe et al., 2010, Chandra et al., 2018).

4. Role of Boundary Conditions and Nonlocality

The hydrodynamic-frame-robust regime critically depends on the interplay between viscous effects, boundary conditions, and system geometry. In two-dimensional electron fluids and graphene, the visibility of key hydrodynamic signatures (e.g., negative nonlocal resistance, vortex formation) depends sensitively on the slip length at the boundary, parametrized by $l_b$. The onset, magnitude, and robustness of these features—particularly in nonlocal voltage and phase correlations—are controlled by the relationship between device size, vorticity diffusion length, and boundary scattering mechanisms (Wagner, 2015, Chandra et al., 2018).

In topologically nontrivial systems, such as Weyl semimetals, "consistent hydrodynamic" models featuring Chern-Simons (topological) terms further enhance the robustness to boundary and disorder effects. Here, nonlocal transport and anomalous Hall currents persist due to the topological invariance of the underlying band structure, leading to spatially asymmetric current and field patterns insensitive to many microscopic details (Gorbar et al., 2018).

5. Theoretical Developments: General Frames, Causality, and Mode Structure

Recent developments in relativistic hydrodynamics have explicitly constructed general-frame theories where the energy-momentum tensor and the constitutive relations incorporate all allowed dissipative and relaxation terms without imposing frame constraints (neither Landau nor Eckart). In such descriptions:

• Additional transient (nonhydrodynamic) degrees of freedom naturally arise and are governed by coupled relaxation equations, with positivity and causality constraints on coefficients derived from the local entropy production (Noronha et al., 2021).
• Frame dependence of higher-order transport coefficients is rendered tractable by focusing on frame-invariant combinations; at the linearized (mode) level, dispersion relations for shear, sound, and diffusion waves are universal functions of frame-invariant transport coefficients, regardless of the nonlinear field redefinitions (Diles et al., 2023).
• In general frames, recasting constitutive equations in the more physical Landau frame necessitates either infinite-order derivative expansions or the introduction of non-fluid (Müller-Israel-Stewart–like) auxiliary variables, with equivalence at the level of hydrodynamic modes but distinctions in their treatment of nonhydrodynamic (gapped) branches (Bhattacharyya et al., 2023).
• “Density Frame” or related constructions yield first-order in time evolution, no “nonhydrodynamic” modes, and improved robustness near the boundary of hydrodynamic applicability, in contrast with second-order or Müller-Israel-Stewart–type schemes where additional parameters influence large-gradient and transient behavior (Bhambure et al., 13 Dec 2024).

In all cases, hydrodynamic predictions become hydrodynamic-frame robust in the sense that the physical (hydro) modes—those controlling the long-wavelength, long-time response—are insensitive to the choice of frame, while nonhydrodynamic modes or frame ambiguities are either gapped away or controllably parametrized.

6. Mathematical Analysis: Well-posedness, Global Solutions, and Biaxial Nematics

The hydrodynamic-frame-robust regime has been rigorously characterized in the mathematical analysis of biaxial nematic liquid crystal flows. The governing equations, coupling the evolution of an orthonormal frame field $p \in SO(3)$ with the Navier-Stokes fluid velocity $v$, admit:

• Global existence and uniqueness: For small initial data, global strong solutions to the Cauchy problem are established, leveraging dissipative energy laws and nonlinear estimates on rotational derivative terms in SO(3) (Feng et al., 8 Aug 2025);
• Well-posedness and weak solution analysis: Local-in-time smooth solutions exist for general data, with blow-up criteria formulated via critical Sobolev or $L^\infty$-type norms on velocity gradients and frame derivatives; in two dimensions, global weak solutions exist aside from finitely many singular times (Li et al., 2022);
• Dissipative structure: The system is endowed with an energy-dissipation law involving kinetic and orientational elastic energies along with SO(3)-covariant dissipation rates, ensuring the robustness of both mathematical and physical modeling frameworks.

The explicit construction and a priori control of rotational derivative terms, as well as tensor decompositions tailored to the SO(3) structure (nine local second-order tensors: symmetric traceless and antisymmetric), are key technical achievements in establishing the hydrodynamic-frame-robust regime for liquids with internal orientational order.

7. Implications, Applications, and Experimental Signatures

The hydrodynamic-frame-robust regime has foundational implications for interpreting experimental data, designing diagnostics, and developing computational models in diverse physical systems:

• Ultracold atomic gases: Crossover in collective modes (zero to first sound, Rayleigh peaks) and dynamic structure factors provide precise probes for distinguishing between non-equilibrium and hydrodynamic regimes (Watabe et al., 2010).
• Graphene and electron hydrodynamics: AC phase correlations, nonlocal resistance minima, and boundary-sensitive vortices distinguish between ballistic and true hydrodynamic transport, robustly identifiable via device geometry and high-frequency drive (Wagner, 2015, Chandra et al., 2018).
• Weyl and topological semimetals: The coexistence of viscous hydrodynamics with topologically protected currents yields nontrivial spatial profiles, negative resistance, and anomalous Hall effects resistant to disorder (Gorbar et al., 2018, Messica et al., 2022).
• Magnon fluids: Experimental detection is guided by the predicted breakdown of the Wiedemann-Franz law for magnons—arising directly due to hydrodynamic frame robustness and differential momentum relaxation channels (Sano et al., 2022).
• High-energy/relativistic fluids: General-frame and causally consistent model constructions enable controlled predictions for relativistic shock-waves and far-from-equilibrium flows, with hydrodynamic modes determined by invariant combinations of transport coefficients (Noronha et al., 2021, Diles et al., 2023, Jaiswal et al., 2023, Bhattacharyya et al., 2023).

In each case, the hydrodynamic-frame-robust regime ensures that macroscopic observables and collective responses are insensitive to ambiguities inherent in non-equilibrium statistical field definitions and that physical predictions remain valid as systems interpolate between kinetic and fully hydrodynamic descriptions. These insights directly inform the modeling, analysis, and interpretation of experiments spanning quantum fluids, electron hydrodynamics, complex anisotropic materials, and relativistic plasmas.