Hydrodynamic Quantum Model

• Hydrodynamic Quantum Model is a framework that describes quantum many-body systems using continuum equations analogous to classical fluid dynamics but modified by quantum effects.
• The model employs moment closure techniques from quantum kinetic theory, such as the Wigner–Boltzmann equation, to derive Euler and Navier–Stokes-like equations incorporating quantum pressure and dispersion.
• Its applications range from superfluidity in cold atomic gases to plasmonics and quantum Hall effects, offering computational efficiency and multiscale insights bridging microscopic and macroscopic phenomena.

The hydrodynamic quantum model refers to a broad class of approaches in which quantum many-body systems are described via collective, macroscopic (long-wavelength) variables – typically density, velocity, and related moments – that evolve according to closed sets of continuum equations analogous to the Euler or Navier–Stokes equations of classical fluid mechanics, but with systematic quantum corrections. This framework underpins the modern understanding of quantum transport, superfluidity, generalized hydrodynamics in integrable systems, and the collective dynamics of cold atomic, electronic, or plasmonic systems.

1. Fundamentals: From Quantum Kinetics to Hydrodynamic Closure

At the core, the hydrodynamic quantum model is constructed by taking low-order moments (density, current, energy, etc.) of an underlying quantum kinetic equation, most commonly the Wigner–Boltzmann equation (for single-particle quantum mechanics or mean-field many-body theory), or the quantum Boltzmann (Uehling–Uhlenbeck) equation for weakly interacting Bose gases. The resulting equations generalize the classical Euler and Navier–Stokes equations by incorporating quantum effects such as the Bohm pressure, higher-order gradient terms, quantum stress, and exchange–correlation potentials (Bose et al., 2015, Jin et al., 2017, Manfredi et al., 2021).

A central feature is the Madelung transformation, which recasts the Schrödinger (or Gross–Pitaevskii) equation into coupled continuity and Hamilton–Jacobi-type equations, rendering quantum mechanics in the form of fluid dynamics with a quantum potential. This gives rise to characteristic quantum phenomena – such as dispersion, superfluidity, and quantum pressure – within a continuum, hydrodynamic formalism (Rashkovskiy, 2013, Manfredi et al., 2021).

2. Canonical Models and Equations

The hydrodynamic quantum model manifests in several key forms:

a. Gross–Pitaevskii Hydrodynamics and Two-Fluid Theory

In weakly interacting Bose–Einstein condensates (BECs), the macroscopic wavefunction Φ(t, r) satisfies a (dissipative) Gross–Pitaevskii equation. A systematic Chapman–Enskog expansion about the full quantum kinetic system yields coupled hydrodynamic equations for the condensate (“superfluid”) and thermal cloud (“normal fluid”), corresponding to Landau’s two-fluid theory:

• Mass exchange:

$$\partial_t n_c + \nabla \cdot (n_c v_c) = -\Gamma,\quad \partial_t n_n + \nabla \cdot (n_n v_n) = +\Gamma,$$

where $n_c, n_n$ are condensate and normal densities, $v_c, v_n$ their velocities, and $\Gamma$ quantifies exchange due to collisions (Jin et al., 2017).

• Momentum:

$$m n_c(\partial_t v_c + v_c \cdot \nabla v_c) + n_c \nabla \mu_c = 0,$$

$$m n_n(\partial_t v_n + v_n \cdot \nabla v_n) + \nabla p_n + n_n \nabla U = - (v_n - v_c) \Gamma,$$

with $\mu_c$ the condensate chemical potential, $p_n$ the pressure of the normal fluid, and $U$ the effective mean-field.

• Navier–Stokes corrections: Dissipation (viscosity, heat flux) for the normal component arises at subleading order, with explicit quantum-modified formulas for the viscous stress tensor and thermal conductivity in terms of the kinetic collisional integrals (Jin et al., 2017).

b. Quantum Hydrodynamic Closure of the Wigner Equation

General quantum hydrodynamic models can be derived by projecting the Wigner–Boltzmann equation via moment methods, yielding a hierarchy of balance laws for particle, momentum, and higher-order moments. In the “all-order” closure, the influence of the quantum potential is retained to arbitrary order in ℏ, with quantum pressure and dispersive effects systematically encoded. The general continuity and momentum equations are:

• Continuity:

$\partial_t n + \nabla \cdot (n \mathbf{u}) = 0$

• Momentum (including quantum pressure tensor):

$$\partial_t (n \mathbf{u}) + \nabla \cdot (n \mathbf{u} \otimes \mathbf{u} + \mathbf{P}) + n \nabla V_\mathrm{KS} + \nabla \cdot \mathbf{P}^Q = 0$$

where $\mathbf{P}$ is the classical pressure, $V_\mathrm{KS}$ the Kohn–Sham potential (for including exchange–correlation), and $\mathbf{P}^Q$ the quantum pressure tensor, expanded in a series in ℏ (Bose et al., 2015).

c. Hydrodynamics of Quantum Integrable Models (Generalized Hydrodynamics)

In 1D integrable systems (e.g., Lieb–Liniger, XXZ chain, Hubbard model), quantum hydrodynamic models at the Euler scale are built for the densities of stable quasiparticles, each evolving via mode-resolved continuity equations:

$$\partial_t \rho_a(\theta, x, t) + \partial_x [v_a^{\rm eff}(\theta, x, t) \rho_a(\theta, x, t)] = 0,$$

where $\rho_a$ is the density of mode $a$ at rapidity $\theta$, and $v_a^{\rm eff}$ is the dressed group velocity determined by Bethe–Ansatz dressing. Diffusive corrections require careful treatment due to long-range correlations and are not always described by a conventional Navier–Stokes form (Hübner, 24 Sep 2025, Ruggiero et al., 2019, Ilievski et al., 2017).

3. Quantum Hydrodynamic Effects: Superfluidity, Quantum Pressure, and Collective Modes

Hydrodynamic quantum models explain a variety of collective quantum phenomena:

• Superfluidity: Appears naturally in models with macroscopic phase coherence, leading to potential (irrotational) flow for the condensate component, with zero viscosity, as in the Euler branch of the two-fluid theory (Jin et al., 2017, Amo et al., 2011).
• Quantum Pressure: The hallmark quantum correction, it accounts for dispersive and nonlocal effects and emerges as the Bohm term $-(\hbar^2/2m) \nabla^2 \sqrt{n} / \sqrt{n}$. This term is essential for phenomena such as soliton formation, interface smoothing (“spill-out” in nanoplasmonics), and stabilizing density gradients.
• Collective Modes: Quantum hydrodynamics captures both acoustic (first sound) and entropy (second sound) modes in Bose gases and superfluids, the full spectrum of hydrodynamic collective excitations in Fermi and Bose systems, and sound/thermal wave propagation in quantum integrable chains (Jin et al., 2017, Sotiriadis, 2016, Mazeliauskas et al., 31 Jan 2025).

4. Mathematical Structure, Regimes, and Closure

a. Systematic Expansions and Closures

The hydrodynamic quantum model is rigorously justified via Chapman–Enskog or Hilbert expansions, in which the quantum kinetic equation is expanded in a small parameter (e.g., the ratio of mean free path to system scale), identifying a hierarchy of transport regimes:

• Euler limit: Collision-dominated, local equilibrium (ballistic) regime.
• Navier–Stokes limit: Next order, capturing dissipation (viscosity, thermal conductivity).
• Higher-order quantum corrections: Arbitrary-order ℏ expansions, nonlocal corrections, and closure schemes via Grad’s moments, retaining global hyperbolicity and well-posedness (Bose et al., 2015, Cai et al., 2012, Hu et al., 2017).

b. Validity and Physical Regimes

Quantum hydrodynamic models are accurate for long-wavelength, collective excitations and at temperatures or densities where quantum dispersion or Pauli/Fermi pressure becomes significant. They are routinely used in:

• Superfluid and BEC dynamics (finite-temperature two-fluid hydrodynamics).
• Plasmonics and electron dynamics in nanoscale systems (QHD for metallic nanoparticles).
• Quantum transport in integrable and nearly-integrable systems.
• Semiconductor quantum devices and low-dimensional systems (Manfredi et al., 2021, Zhou et al., 2021, Hu et al., 2017).

Limitations include breakdown beyond the mean-field regime, in strongly disparate velocity distributions, or where full quantum decoherence, entanglement, or higher-order quantum correlations dominate.

5. Extensions, Applications, and Experimental Relevance

Hydrodynamic quantum models have far-ranging applications:

• Bose gases: Quantitative modeling of collective modes, vortex dynamics, and nonequilibrium evolution in trapped gases, accurately capturing Landau damping, sound propagation, and finite-temperature effects (Jin et al., 2017).
• Plasmonics: Efficient large-scale modeling of quantum-size effects, spill-out, and nonlocal response in nanostructured metals, with semiquantitative agreement with time-dependent density functional theory benchmarks (Ciracì et al., 2016, Zhou et al., 2021).
• Quantum Hall fluids: Effective hydrodynamics for the fractional quantum Hall effect, including quantum pressure, Chern–Simons terms, Hall viscosity, and quantized conductance (Abanov, 2012).
• Integrable models: Emergence of ballistic and diffusive hydrodynamic scaling in 1D quantum gases, XXZ/Hubbard chains, and cold atom experiments; discovery of “hydrodynamic attractors” and out-of-equilibrium generalizations (Ruggiero et al., 2019, Kharkov et al., 2021, Mazeliauskas et al., 31 Jan 2025).
• Non-equilibrium states and solitons: Formation and evolution of quantum hydrodynamic solitons, vortex nucleation, and turbulence in polariton and atomic condensates (Amo et al., 2011, Sotiriadis, 2016).

6. Computational and Theoretical Advantages

Quantum hydrodynamic models provide substantial computational efficiency relative to fully microscopic quantum simulations, enabling simulation of large-scale systems (nano-objects, condensates, cold gases) with relatively low computational cost. The fluid description naturally incorporates spin, multi-component, electromagnetic, and relativistic corrections, and is amenable to well-posed boundary value and initial value problems, with established theorems guaranteeing existence, uniqueness, and stability under physically realistic conditions (Manfredi et al., 2021, Hu et al., 2017).

A plausible implication of this unified framework is the potential to connect quantum effective field theory, kinetic theory, and macroscopic phenomenology in a controlled, multiscale way, enabling both analytic intuition and quantitative predictions across scales.

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References:

• Quantum hydrodynamic approximations to the finite temperature trapped Bose gases (Jin et al., 2017)
• Construction of a more complete quantum fluid model from Wigner-Boltzmann Equation with all higher order quantum corrections (Bose et al., 2015)
• Fluid descriptions of quantum plasmas (Manfredi et al., 2021)
• Eulerian and Newtonian dynamics of quantum particles (Rashkovskiy, 2013)
• Quantum hydrodynamic modeling of edge modes in chiral Berry plasmons (Zhang et al., 2017)
• Discovering hydrodynamic equations of many-body quantum systems (Kharkov et al., 2021)
• Hydrodynamic theory of quantum fluctuating superconductivity (Davison et al., 2016)
• Polariton superfluids reveal quantum hydrodynamic solitons (Amo et al., 2011)
• On the effective hydrodynamics of FQHE (Abanov, 2012)
• Quantum Generalized Hydrodynamics (Ruggiero et al., 2019)
• Stability and semi-classical limit in a semiconductor full quantum hydrodynamic model with non-flat doping profile (Hu et al., 2017)
• Calibrating quantum hydrodynamic model for noble metals in nanoplasmonics (Zhou et al., 2021)
• Quantum Hydrodynamic Model by Moment Closure of Wigner Equation (Cai et al., 2012)
• Equilibration in one-dimensional quantum hydrodynamic systems (Sotiriadis, 2016)
• On the Hydrodynamic Approximation of Quantum Integrable Models (Hübner, 24 Sep 2025)
• Hydrodynamic attractor in periodically driven ultracold quantum gases (Mazeliauskas et al., 31 Jan 2025)
• Quantum Hydrodynamic Theory for Plasmonics: Impact of the Electron Density Tail (Ciracì et al., 2016)