Generalized Hydrodynamics Framework

• Generalized Hydrodynamics (GHD) is a framework describing large-scale out-of-equilibrium dynamics of integrable systems via the phase-space distribution of quasiparticles.
• It incorporates an infinite set of local conservation laws and uses state-dependent effective velocities, derived from dressing techniques and Bethe ansatz equations, to close hydrodynamic equations.
• Validated in both quantum and classical systems, GHD accurately predicts experimental outcomes in cold atom setups and explains shock-free evolution in many-body dynamics.

Generalized Hydrodynamics (GHD) is a framework for describing the large-scale out-of-equilibrium dynamics of integrable quantum and classical many-body systems in one spatial dimension. GHD generalizes conventional hydrodynamics by incorporating the infinite set of local conservation laws characteristic of integrable models. The core variable is the phase-space density of quasiparticles (or solitons), whose locally defined distribution encodes the occupation of Bethe rapidities or spectral parameters. The evolution of this density obeys a closed set of coupled Euler-type continuity equations with state-dependent effective (dressed) velocities. GHD has been rigorously validated in both quantum and classical systems and underlies the macroscopic behavior of cold atom experiments on 1D Bose gases, classical soliton gases, and other integrable platforms (Schemmer et al., 2018, Bonnemain et al., 2022, Doyon et al., 2023).

1. Foundational Equations and Thermodynamic Structure

In any integrable system, there exists an infinite family of local conservation laws, each associated with a charge whose density and current can be expressed in terms of underlying quasiparticle distributions. The central objects in GHD are:

• The phase-space (rapidity) density $\rho_{\rm p}(x,t;\theta)$ of quasiparticles with spectral parameter $\theta$ at position $x$ and time $t$.
• The state (available) density $\rho_{\rm s}(x,t;\theta)$, which incorporates the effect of interactions via the Bethe–Takahashi (or Bethe–Yang) equations and is related to the dressing of bare quantities.
• The filling function $$n(x,t;\theta) = \rho_{\rm p}(x,t;\theta)/\rho_{\rm s}(x,t;\theta)$$, a normalized occupation factor $0\le n \le 1$.

The time evolution at Euler scale is governed by the continuity equation: $$\partial_t \rho_{\rm p}(x,t;\theta) + \partial_x\bigl(v^{\rm eff}_{[n(x,t)]}(\theta) \, \rho_{\rm p}(x,t;\theta)\bigr) = 0,$$ where $v^{\rm eff}_{[n(x,t)]}(\theta)$ is the state-dependent effective velocity. All conserved charges and currents are functionals of $\rho_{\rm p}$ and are obtained via linear functionals with dressed one-particle eigenvalues (Doyon et al., 2017, Doyon et al., 2023).

At equilibrium, as realized in the thermodynamic Bethe Ansatz (TBA), the pseudoenergy $\epsilon(\theta)$ solves nonlinear integral equations that define the Yang–Yang entropy and other thermodynamic quantities. Any bare observable $h(\theta)$ is "dressed" by the integral equation: $$h^{\rm dr}(\theta) = h(\theta) + \int \frac{d\theta'}{2\pi} \phi(\theta-\theta') n(\theta') h^{\rm dr}(\theta'),$$ where $\phi(\theta-\theta')$ is the two-body scattering kernel, depending on the specific model (e.g., $\phi(\theta) = 2\arctan((\theta-\theta')/g)$ for the Lieb–Liniger gas) (Schemmer et al., 2018, Bonnemain et al., 2022).

2. Effective (Dressed) Velocity and Hydrodynamic Closure

The key dynamical object in GHD is the effective velocity of quasiparticles, encoding the state-dependent renormalization of their propagation due to interactions: $$v^{\rm eff}(\theta) = \frac{(E')^{\rm dr}(\theta)}{(p')^{\rm dr}(\theta)},$$ where $E'(\theta)$ and $p'(\theta)$ are the derivatives of the bare energy and momentum, and the dressing is performed as above. The effective velocity equation, in kinetic theory notation, reads: $$v^{\rm eff}(\theta) = v(\theta) + \int d\theta' \, \rho_{\rm p}(\theta') \Delta(\theta-\theta') [v^{\rm eff}(\theta) - v^{\rm eff}(\theta')],$$ where $v(\theta)$ is the bare velocity and $\Delta(\theta-\theta')$ is the two-body scattering shift (Schemmer et al., 2018).

This structure ensures the closure of the hydrodynamic equations for infinitely many conserved densities: the evolution of $\rho_{\rm p}$ is governed solely by local data and the dressing, without ad hoc macroscopic closure assumptions. The entropy and other thermodynamic quantities evolve consistently due to the underlying TBA structure (Doyon et al., 2016).

3. Extensions: External Potentials, Non-Homogeneous Interactions, and Impurities

GHD is not restricted to homogeneous Hamiltonians. In the presence of external potentials $V(x)$ or slowly varying space-time couplings $g(x,t)$, the continuity equation generalizes to: $$\partial_t \rho_{\rm p} + \partial_x(v^{\rm eff}\rho_{\rm p}) + (\partial_x V(x)/m) \, \partial_\theta \rho_{\rm p} = 0,$$ with further force terms when interactions become inhomogeneous: $$\partial_t\rho + \partial_x(v^{\rm eff}\rho) + \partial_\lambda\left( \frac{\partial_t g f^{\rm dr} + \partial_x g \Lambda^{\rm dr}}{(p')^{\rm dr}} \rho \right) = 0,$$ where $f$ and $\Lambda$ are force densities deriving from the explicit $g$-dependence of the Hamiltonian (Bastianello et al., 2019).

For models with point-like integrable impurities, a collision integral appears localized at the impurity position: $$\partial_t \rho_j(\lambda,x,t) + \partial_x\left[v^{\rm eff}_j(\lambda,x,t) \rho_j(\lambda,x,t)\right] = \delta(x) \mathcal{I}_j(\lambda,t),$$ where $\mathcal{I}_j$ encodes exact reflection and transmission, dressed by the local state (Rylands et al., 2023).

4. Soliton Gases, Classical Chains, and Statistical Mechanical Interpretation

GHD encompasses not only quantum integrable systems but also classical soliton gases (e.g., KdV and Boussinesq equations) and harmonic chains. In these settings, the quasiparticle densities represent soliton or mode occupation and dressing incorporates statistical effects from many-body scattering. The kinetic equations are structurally identical, and thermodynamics derives from a generalized Gibbs ensemble imposing conservation of the infinite set of charges: $$\rho \propto \exp\left( -\sum_n \beta_n Q_n \right),$$ with free energy, currents, and correlation matrices expressible in terms of the filling and dressed charges. This statistical mechanical perspective unifies quantum and classical GHD (Bonnemain et al., 2022, Bonnemain et al., 2024, Pandey et al., 2024).

5. Mathematical Structure, Integrability, and Shock-Free Dynamics

GHD has a Hamiltonian structure: the phase-space density $\rho(x,\theta)$ is the dynamical field, and flows are generated by a Poisson bracket characterized by a dressing operation associated with a symmetric interaction kernel $\psi(x,\theta;x',\theta')$: $$\{F,G\} = \int dx\,d\theta\, \frac{n(x,\theta)}{2\pi} \left( F'_x (G'_\theta)^{\rm dr} - G'_x (F'_\theta)^{\rm dr} \right),$$ for arbitrary functionals $F,G$ of $\rho$. The total energy acts as the Hamiltonian, and GHD can be viewed as a $2+1$-dimensional classical field theory (Bonnemain et al., 2024).

Remarkably, for the Lieb–Liniger and related models, rigorous results establish that the Euler-scale GHD evolution is globally smooth: no shocks (gradient catastrophes) can appear at any time. For discontinuous initial data, GHD selects a unique entropy-preserving weak solution, unlike conventional Euler hydrodynamics where ambiguities and non-uniqueness arise (Hübner et al., 2024, Doyon et al., 2017). The shock-free property reflects the nonlinearly degenerate structure of GHD, ultimately tied to the infinite number of conservation laws.

6. Numerical Methods and Analytical Solutions

Numerical solution of the GHD equations leverages discretization of rapidity and space, with time integration performed by finite-difference, upwind, or characteristic-following schemes. Dressing is typically implemented via linear solvers (e.g., matrix inversion for Fredholm equations). Efficient approaches include geometric formulations where GHD maps to advection in a state-dependent metric, with solution via nonlinear Fredholm-type integral equations, and Banach fixed-point algorithms providing rapid convergence and global existence (Doyon et al., 2017, Hübner et al., 2024). Public frameworks such as iFluid encapsulate these algorithms for a wide range of integrable models (Møller et al., 2020).

7. Physical Validation, Experimental Realizations, and Role in Current Research

GHD is extensively validated in ultracold atom experiments on 1D Bose gases. The time evolution of the in situ density profile after quenches of confinement or potential protocol changes is predicted quantitatively over the full temperature and interaction range (Schemmer et al., 2018, Bouchoule et al., 2021). The infinite-component GHD equations correctly describe scenarios where conventional hydrodynamics fails, most notably for highly nonthermal initial distributions (e.g., double-well to harmonic trap quenches and "quantum Newton's cradle" experiments), where CHD develops spurious shocks or fails to reproduce rapidly evolving multi-modal rapidity distributions (Doyon et al., 2017). GHD also provides the theoretical underpinning for describing ballistic and diffusive transport, relaxation to non-equilibrium steady states, and statistical properties of fluctuations and large deviation functions in integrable systems (Doyon et al., 2023, Watson et al., 2022).

The framework's extension to fluctuating hydrodynamics, the analysis of prethermalization phenomena, and the interplay with effective field theories (e.g., the emergence of Luttinger liquid theory as the quantization of zero-entropy GHD) represent active research frontiers (Ruggiero et al., 2019, Møller et al., 2022, Urilyon et al., 27 May 2025). Analytical control over correlation functions at and beyond the Euler scale, inclusion of integrability-breaking perturbations, and the unified description of quantum and classical collective dynamics position GHD as a central theoretical structure in nonequilibrium statistical mechanics.