Hydrodynamic Spectral Correlations

• Hydrodynamic spectral correlation functions are defined as time and space Fourier transforms of fluctuations in conserved fields, revealing resonant features such as sound and heat modes.
• They encode the propagation and decay of collective excitations through resonant poles and non-analytic structures, linking phenomena like Brillouin doublets and Rayleigh peaks to transport coefficients.
• They bridge theoretical models and experimental data, enabling quantification of viscosities, diffusivities, and elastic properties via analysis of scattering spectra.

Hydrodynamic spectral correlation functions characterize the frequency-resolved spatiotemporal fluctuations of conserved densities and currents in classical, quantum, or active fluids, as well as in crystalline solids. These functions encode the propagation and decay of hydrodynamic modes—such as sound, heat, and shear—through their resonant poles and non-analytic structures in the complex frequency plane. They provide a unifying framework connecting transport coefficients, mode structure, and fluctuation-response relations across equilibrium and non-equilibrium states, and are fundamental for the interpretation of experimental scattering spectra and the development of microscopic kinetic and field-theoretic theories.

1. Core Definitions: Time-Dependent Correlation and Spectral Functions

Hydrodynamic spectral correlation functions are defined as time and space Fourier transforms of dynamical correlation functions of hydrodynamic fields, such as density, momentum, or energy. For a generic conserved field $\phi(\mathbf{r},t)$, the fundamental two-point correlation in Fourier space is

$$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$

For mass, momentum, and energy, one defines:

• Dynamic structure factor (density fluctuations):

$$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$

• Current spectral function (e.g., longitudinal):

$$C_l(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle j_\parallel(q,t) j_\parallel(-q,0) \rangle,$$

where $j_\parallel$ is the Fourier transform of the longitudinal momentum density.

The peaks, pole structure, and analytic features of these spectra directly encode the collective excitation spectrum and transport properties of the system (Vollmayr-Lee et al., 2010, Mabillard et al., 2023, Mabillard et al., 2023, Mabillard et al., 2023).

2. Spectral Functions in Fluctuating Hydrodynamics

Hydrodynamic spectral correlation functions are governed by the linearized fluctuating hydrodynamic equations, which express the conservation of mass, momentum, and (granular/thermal) energy with both deterministic and stochastic terms. For granular or molecular fluids,

$$\begin{cases} \partial_t\,\delta n(q,t) = -i q n_0 u(q,t),\ \partial_t\,u(q,t) = \ldots - \nu_l q^2 u(q,t) + \xi_l(q,t), \ \partial_t\,\delta T(q,t) = - D_T q^2 \delta T(q,t) - \Gamma_0 ( \ldots ) + \theta(q,t), \end{cases}$$

where $\xi_l, \theta$ are Gaussian noise terms consistent with the fluctuation-dissipation theorem, and $\nu_l, D_T$ are longitudinal kinematic viscosity and thermal diffusivity, respectively (Vollmayr-Lee et al., 2010, Mabillard et al., 2023).

The solutions yield closed-form or mode-resolved expressions for the spectra, e.g. for density fluctuations (dynamic structure factor $S(q,\omega)$) or longitudinal currents ($C_l(q,\omega)$), in terms of the hydrodynamic parameters and the system's equation of state (Vollmayr-Lee et al., 2010, Mabillard et al., 2023).

3. Resonant Structure and Mode Content

Hydrodynamic spectral functions generically exhibit a set of resonances corresponding to collective modes:

• Rayleigh peak: Centered at $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$0, with width $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$1, associated with (thermal or granular) diffusion.
• Brillouin doublet: Peaks at $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$2, with widths $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$3, corresponding to damped sound (acoustic) modes.
• Shear (transverse) modes: In fluids, appear as a central Lorentzian at $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$4, width $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$5 (shear viscosity). In crystals, these split to propagating transverse phonons at $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$6 (Mabillard et al., 2023, Mabillard et al., 2023).

Poles of $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$7 and related spectra are functions of $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$8, whose real and imaginary parts give propagation velocities and damping rates, directly linked to transport coefficients (viscosities, diffusivities, sound speeds) (Mabillard et al., 2023, Mabillard et al., 2023, Vollmayr-Lee et al., 2010).

In non-equilibrium or driven systems (e.g., driven granular fluids), the mode structure also reflects explicit inelastic loss and external driving, producing finite relaxation rates for temperature modes at $$S_{\phi\phi}(\mathbf{q},\omega) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t} \langle \delta\phi(\mathbf{q},t) \delta\phi(-\mathbf{q},0) \rangle.$$9 (Vollmayr-Lee et al., 2010).

4. Extraction of Transport Coefficients and Comparison to Kinetics

Transport coefficients, such as shear and bulk viscosities, and thermal diffusivity, can be extracted either from the widths of peaks in the hydrodynamic spectral functions or from time-integrals of real-time correlation functions via Green–Kubo or Einstein–Helfand formulas: $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$0 where $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$1 is the relevant Helfand moment (Mabillard et al., 2023, Mabillard et al., 2023).

Molecular-dynamics studies have established the quantitative equivalence between transport values obtained from spectral pole extraction and those calculated by Helfand moments, validating the fluctuating hydrodynamics prescriptions even for hard-sphere fluids and perfect crystals (Mabillard et al., 2023, Mabillard et al., 2023). In granular systems, simulation-extracted values agree to within 10–20% with kinetic theory predictions (Vollmayr-Lee et al., 2010).

5. Regimes and Phenomenology: Fluids, Granular Media, and Crystals

Spectral correlation functions capture a range of physical regimes:

• Normal fluids: Exhibit five hydrodynamic modes (two sound, one heat, two shear diffusion), reflected as Rayleigh and Brillouin structures in $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$2 and Lorentzian shear peaks in transverse spectra (Mabillard et al., 2023).
• Granular fluids: Additional finite-rate cooling produces central “inelastic” Lorentzian peaks associated with temperature relaxation (finite width even at $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$3), and the crossover between “inelastic-cooling–dominated” and “diffusive” regimes is controlled by $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$4 (Vollmayr-Lee et al., 2010).
• Crystals: Continuous translational symmetry breaking yields a seven-mode structure—two longitudinal (Brillouin), one central (heat diffusion), and four transverse (propagating transverse phonons)—observable as distinct resonances in $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$5 and momentum-density spectra (Mabillard et al., 2023, Mabillard et al., 2023).

These resonances are quantitatively described by analytic hydrodynamic expressions using independently defined elastic and transport coefficients, and have been rigorously matched to simulation data (Mabillard et al., 2023, Mabillard et al., 2023, Mabillard et al., 2023, Vollmayr-Lee et al., 2010).

6. Influence of Nonequilibrium and System-Specific Physics

Hydrodynamic spectral correlation functions adapt to system-specific features:

• Driven granular media: Incorporate inelastic cooling, external driving, and modified noise correlators, resulting in nontrivial crossovers in temperature-relaxation mechanisms and the emergence of “finite-rate” decay regimes not encountered in elastic fluids (Vollmayr-Lee et al., 2010).
• Crystals: Require projection of elastic and viscous tensors onto crystal symmetry axes to fully characterize the splitting and attenuation of phonon branches (Mabillard et al., 2023, Mabillard et al., 2023).
• Anisotropy or symmetry-breaking: Affects the number and dispersion of hydrodynamic modes, leading to observable multi-peak structure in spectral functions reflecting broken symmetries (Mabillard et al., 2023).
• Quantum or relativistic hydrodynamics: Analysis of the analytic structure (e.g., poles, branch cuts) of spectral functions provides direct information about microscopic causality, stability, and universality in the underlying theories (Abbasi et al., 2022).

7. Practical Role in Theory and Experiment

Spectral correlation functions bridge first-principles statistical-mechanical theory, numerical simulations, and experiment:

• Experimental observation: Dynamic light or neutron scattering spectra directly measure $$S(q,\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t}\, \langle \delta n(q,t) \delta n(-q,0) \rangle.$$6, from which sound speeds, viscosities, and diffusivities are extracted by fitting to hydrodynamic lineshapes.
• Microscopic hydrodynamic validation: Excellent quantitative agreement between hydrodynamic theory (parameterized solely by measured transport and elastic coefficients), molecular dynamics, and kinetic theory demonstrates robust applicability of local-equilibrium hydrodynamics for both fluids and solids (Mabillard et al., 2023, Mabillard et al., 2023, Vollmayr-Lee et al., 2010).
• Role in non-equilibrium systems: Ability to incorporate driving, dissipation, and anomalous noise structures extends the reach of hydrodynamic spectral analysis to driven steady states and complex media.

Hydrodynamic spectral correlation functions thus constitute the essential core of the statistical mechanics of extended systems, providing rigorous quantitative links between macroscopic transport, microscopic dynamics, and collective fluctuation phenomena (Vollmayr-Lee et al., 2010, Mabillard et al., 2023, Mabillard et al., 2023, Mabillard et al., 2023).