Hydrodynamic Complex Linear Momenta

• Hydrodynamic complex linear momenta are defined as momentum-like observables derived from spatial Fourier transforms that capture the analytic structure and dynamics in fluid systems.
• They are central to understanding spectral instabilities and the convergence of gradient expansions through branch point and pseudospectral analyses.
• Their applications span quantum fluids, mesoscopic hydrodynamics, and cold quantum gases, where they influence transport properties and stability.

Hydrodynamic complex linear momenta describe the spectrum, dynamics, and analytic structure of momentum-like observables in hydrodynamic systems when considered in the complexified momentum (or wavenumber) plane. These quantities arise in contexts ranging from the spectral theory of kinetic and quantum-classical models to the study of diffusion, absorption lengths, and non-analyticity in strongly coupled plasmas and quantum fluids. Complex linear momenta play a central role in hydrodynamic gradient expansions, control the convergence and stability properties of hydrodynamic modes, and encode the onset of nonhydrodynamic physics through singularities and branch points in their analytic structure.

1. Spectral Theory and Complex Momentum Modes

The formulation of hydrodynamic modes in the complex momentum plane originates from linear response theory, where spatial Fourier transforms convert conservation laws into algebraic dispersion-relations involving $\omega$ and $k$ (frequency and wavenumber). For a given frequency, the hydrodynamic problem reduces to seeking eigenvalues $k(\omega)\in\mathbb{C}$ such that the corresponding field perturbations both satisfy infalling regularity at a black brane horizon (in AdS/CFT) or appropriate microscopic boundary conditions, and vanish at infinity or the system boundary. In the context of holographic diffusion, the longitudinal Maxwell sector admits such complex linear momenta, which are interpreted holographically as the poles governing the spatial absorption length of conserved currents in the boundary theory (Garcia-Fariña et al., 2 Feb 2026).

For frequencies much smaller than the thermal or microscopic scale, hydrodynamics predicts that the dominant complex linear momenta are given by a Puiseux expansion:

$$k_{\pm}(\omega) = \pm \sqrt{\frac{i\,\omega}{D} + \mathcal{O}(\omega^{3/2})}$$

where $D$ is the diffusion constant. The presence of a square-root branch point at $\omega=0$ indicates an exceptional point where the two momentum roots collide, leading to enhanced spectral instability.

2. Gradient Expansion, Branch Points, and Critical Momenta

Hydrodynamic gradient expansions express constitutive relations (e.g., the stress tensor or current) as Taylor series in $k$ about $k=0$, with each transport coefficient associated with higher-order derivatives. However, the radius of convergence of such expansions is generically set by non-analyticities (branch points) in the complex $k$-plane, arising at the collision of hydrodynamic modes with nonhydrodynamic excitations (Heller et al., 2020). For relativistic viscous fluids, exact or model-specific dispersion relations, such as those in Müller–Israel–Stewart (MIS) theory,

$$\omega_{\rm shear}(k) = -i D k^2 - i (D^2\tau_\pi - \tfrac{\theta_1}{2sT}) k^4 + \mathcal{O}(k^6),$$

possess a square-root singularity at

$k_*^2 = \frac{1}{4 D \tau_\pi},$

defining the critical momentum $k_c$. The large-order growth of the gradient series coefficients is set by this branch point; the expansion converges when all Fourier support lies within $|k| < k_c$.

A similar spectral structure is established for kinetic models such as the linearized Boltzmann BGK operator. For each hydrodynamic branch (shear, diffusion, acoustic), the spectrum exhibits critical wavenumbers $k_{\rm crit}^{(\ell)}$ beyond which no isolated hydrodynamic eigenvalues persist, and the system's response transitions from fluid-like to kinetic (Kogelbauer et al., 2023).

3. Pseudospectra, Instability, and Exceptional Points

The spectral problem for complex momentum modes in non-self-adjoint operators is fundamentally non-normal, implying that eigenvalues alone do not fully characterize the system's stability or response to perturbations. The appropriate diagnostic is the $\varepsilon$–pseudospectrum:

$$\sigma_\varepsilon(L) = \{ z \in \mathbb{C} \mid \| (L - z I)^{-1} \| > 1/\varepsilon \},$$

where the operator norm is computed in the physical energy metric (Garcia-Fariña et al., 2 Feb 2026).

Hydrodynamic complex linear momenta, particularly those associated with diffusion, display enhanced pseudospectral instability in the vicinity of branch points (exceptional points). Specifically, for $k_{\pm}(\omega)$ near $\omega=0$, small perturbations $\varepsilon$ can induce eigenvalue shifts scaling as $\varepsilon^{1/2}$, with the pseudospectral condition number diverging as $1/\sqrt{\omega}$. This signals an extreme sensitivity: absorption lengths for diffusive transport modes become highly susceptible to microscopic deformations, and conventional analytic perturbation theory fails. This phenomenon parallels the behavior near second-order exceptional points in non-Hermitian spectral theory.

This enhanced instability of complex hydrodynamic momenta is uniquely a feature of the "fixed $\omega$, variable $k$" (complex-momentum) formulation; it is absent in the "fixed $k$, quasinormal frequency" approach, where the hydrodynamic QNM frequency $\omega(k) \approx -i D k^2$ exhibits much greater spectral stability.

4. Analytic Structure: Univalence, Stability, and Causality

The analytic properties of hydrodynamic dispersion relations in the complex momentum plane dictate not only the convergence of the gradient expansion, but also the single- or multi-valuedness ("univalence") of the frequency as a function of $k^2$ (Heydari et al., 2024). This has implications for identifying pathological behaviors such as mode self-intersections or cusps.

Analysis of high-order hydrodynamic models (e.g., MIS, BDNK) reveals that, generically, the hydro series is locally but not globally univalent in $k$. Shear channels are often univalent everywhere within the convergence radius, aligning with regions of stability (Im $\omega \le 0$) and causality ($|{\rm Re}\, \omega/{\rm Re}\, k| \le 1$). In contrast, sound channels may lose univalence or even analytic convergence at unexpectedly low $k$ due to multiple branch-point collisions. In these cases, the region where the Bieberbach bounds $|a_n| \leq n$ (single-valuedness of expansion coefficients) are satisfied provides nontrivial constraints on transport coefficients beyond those from stability or causality requirements.

This analytic perspective unifies gradient expansion, mode stability, and the physical requirement that hydrodynamic dispersion relations propagate without unphysical doubling.

5. Complex Linear Momentum in Quantum-Classical and Mesoscopic Hydrodynamics

Beyond classical and relativistic fluids, complex linear momenta arise in quantum hydrodynamic and mixed quantum–classical settings. In Madelung hydrodynamics, the local complex momentum density is given by

$$P(q,t) = \chi^*(-i\nabla)\chi = M D u - i \frac{1}{2} \nabla D,$$

where $D$ is the density, $u$ the velocity, and the imaginary part encodes quantum pressure contributions (Gay-Balmaz et al., 2023). The total complex linear momentum, integrating both real hydrodynamic and imaginary quantum-gradient parts, is a conserved quantity under the Hamiltonian structure.

In mesoscopic and active complex fluids, linear momenta at the scale of coarse-grained "cells" are characterized by Irving–Kirkwood-type averages over neighbor distributions. The mesoscopic stress tensor is expressed as

$$\mathbf{G}(\mathbf{r}) = \frac{Z \rho_1}{2} \langle \boldsymbol{\epsilon} \otimes \mathbf{F} \rangle_2,$$

where $\boldsymbol{\epsilon}$ is the cell–cell vector, $\mathbf{F}$ the net momentum flux, and $Z, \rho_1$ are the local coordination and density (Fruleux et al., 2014). This formalism generalizes the construction of linear momentum balance to systems with unknown microscale physics, encompassing active, polar, or living materials.

6. Hydrodynamic Complex Linear Momenta in Cold Quantum Gases

In hydrodynamic approaches to cold Bose gases, the distinction between fast "inner" quantum modes (e.g., local many-body wavefunctions with nonzero inner momentum) and slow hydrodynamic variables yields effective equations containing additional complex momentum densities. The continuity equation features an extra current

$$P_{\rm inner}(x, t) = \rho(x,t) \frac{p_{\rm inner}(x,t)}{m}$$

from inner quantum excitations, and the full momentum equation includes momentum-exchange couplings between slow and fast modes (Pergamenshchik, 2024). Nonzero $p_{\rm inner}$ enables qualitatively novel behaviors such as soliton propagation and density-current asymmetries, inaccessible in standard local-density approximations.

7. Force Moments and Complex Fluid Environments

The hydrodynamic force moments of embedded particles (including linear momentum and stresslets) in complex, weakly non-Newtonian fluids are governed by generalized reciprocal theorems (Elfring, 2017). This enables the extraction of momentum transfer—including non-Newtonian corrections—without explicit solution for the disturbance flow, provided known Newtonian resistance operators are available. The coupling of complex momenta with non-Newtonian stresses leads to modifications of swimming dynamics, stress transmission, and rotational behaviors in active suspensions and rheologically complex media.

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Hydrodynamic complex linear momenta unify the analytic, spectral, and transport-theoretic aspects of fluid, kinetic, and quantum hydrodynamic systems. They provide a rigorous language for describing the limits of hydrodynamic applicability, characterize the exceptional-point phenomena underpinning stability and instability, and facilitate the extension of hydrodynamics to mesoscopic, quantum-classical, and complex fluid environments. The study of their branch point structure, pseudospectral properties, and their conservation laws continues to inform both foundational and applied hydrodynamics across scales and interaction regimes (Garcia-Fariña et al., 2 Feb 2026, Heller et al., 2020, Gay-Balmaz et al., 2023, Fruleux et al., 2014, Heydari et al., 2024, Pergamenshchik, 2024, Kogelbauer et al., 2023, Elfring, 2017).