The Euler-scale equations of generalised hydrodynamics (GHD) - the hydrodynamic theory of one-dimensional integrable systems, confirmed experimentally in cold-atom gases - have never been derived from the microscopic dynamics of an interacting quantum model: all available derivations assume local relaxation to generalised Gibbs ensembles. Doyon and Hübner proposed an ab initio programme starting instead from the Bethe wave functions of the repulsive Lieb-Liniger model, identifying the GHD quasi-particles as wave packets that evolve under a classical particle model with two-body scattering shifts. They presented four routes to the GHD equation for averaged conserved densities, each with explicitly identified gaps. The central bottleneck is the sum over the N!N! permutations in the Bethe wave function: the stationary-phase and phase-linearisation approximations control each permutation sector separately, but not their coherent sum in the large-scale, finite-density limit; the working assumption that oscillatory phases force cancellation except in diagonal sectors is unjustified and contradicts the emergent fermionic statistics of the quasi-particles; a large-deviation-type framework for oscillatory integrals encoding particle statistics is missing; and neglecting the off-diagonal sectors visibly breaks normalisation and unitarity through a lost macroscopic measure. On the operator route, the exact normal-ordered fixed-point identity for the quasi-particle height field still awaits proofs that normal ordering is negligible at large scales and that large-scale observables factorise. The problem - restated as open in the published 2026 version and confirmed open by the field's reviews - is to complete such a derivation with controlled approximations, equivalently to supply the missing asymptotic control of the permutation sums or of the operator-route steps.

Background

Quantum integrable models in one spatial dimension, such as the Lieb-Liniger gas of NN bosons on a line repelling each other by contact forces of strength c>0c>0 (Lieb and Liniger, 1963), possess an infinite family of extensive conserved quantities. This prevents ordinary thermalisation and makes conventional hydrodynamics inapplicable. The correct Euler-scale theory is generalised hydrodynamics (GHD), introduced in 2016 by Castro-Alvaredo, Doyon and Yoshimura and by Bertini, Collura, De Nardis and Fagotti. Its central object is the spectral phase-space density ρp(t,x,θ)\rho_{\rm p}(t,x,\theta): the quantity dθ dx ρp\mathrm{d}\theta\,\mathrm{d}x\,\rho_{\rm p} counts quasi-particles whose rapidity θ\theta (a convenient parametrisation of momentum) lies in [θ,θ+dθ][\theta,\theta+\mathrm{d}\theta] within a mesoscopic fluid cell around (t,x)(t,x). It satisfies a kinetic-looking conservation law

∂tρp(t,x,θ)+∂x(veff(t,x,θ) ρp(t,x,θ))=0,\partial_{t}\rho_{\rm p}(t,x,\theta)+\partial_{x}\big(v^{\rm eff}(t,x,\theta)\,\rho_{\rm p}(t,x,\theta)\big)=0,

where the effective velocity veffv^{\rm eff} is a nonlinear functional of the local state, fixed by the integral equation

veff(θ)=v(θ)+∫dα  φ(θ−α) ρp(α) (veff(α)−veff(θ)).v^{\rm eff}(\theta)=v(\theta)+\int \mathrm{d}\alpha\;\varphi(\theta-\alpha)\,\rho_{\rm p}(\alpha)\,\big(v^{\rm eff}(\alpha)-v^{\rm eff}(\theta)\big).

Here v(θ)v(\theta) is the bare one-particle velocity (v(θ)=θv(\theta)=\theta for Lieb-Liniger) and φ(θ)=ϕ′(θ)\varphi(\theta)=\phi'(\theta) is the two-body semi-classical scattering shift, the derivative of the exact two-particle scattering phase ϕ(θ)=2arctan⁡(θ/c)\phi(\theta)=2\arctan(\theta/c).

All existing derivations of these equations in quantum models use the hydrodynamic assumption: every fluid cell is assumed to relax locally to a generalised Gibbs ensemble (GGE) - the entropy-maximising state given the infinitely many conserved quantities - and the thermodynamic Bethe ansatz (TBA; the finite-volume Bethe quantisation conditions followed by the thermodynamic limit N∼ℓ→∞N\sim\ell\to\infty on a system of size ℓ\ell) supplies the expectation values of densities and currents. Such derivations do not explain the kinetic picture behind the equation, rest on the uncontrolled local-relaxation assumption, and contain logical irritants such as periodic fluid cells and the spurious momentum quantisation they introduce.

Doyon and Hübner instead proposed an ab initio programme starting from the wave functions themselves. The coordinate Bethe ansatz gives the exact Lieb-Liniger eigenfunctions as

ψ(x∣θ)=cN s(x)∑σ∈SN(−1)∣σ∣ eiΦ(xσ,θ),\psi(\mathbf{x}|\boldsymbol{\theta})=c_{N}\,s(\mathbf{x})\sum_{\sigma\in\mathcal{S}_{N}}(-1)^{|\sigma|}\,e^{i\Phi(\mathbf{x}_{\sigma},\boldsymbol{\theta})},

an (in the rapidities θ\boldsymbol{\theta}) antisymmetrised sum over the N!N! permutations σ\sigma of plane waves dressed by the Bethe phase Φ\Phi, which accumulates the pairwise scattering phases ϕ(θi−θj)\phi(\theta_i-\theta_j). The programme builds long-wavelength, finite-density initial states as superpositions of such eigenfunctions localised around wave-packet phase-space positions (a gas of wave packets) and follows the wave function in the scaling limit ℓ→∞\ell\to\infty, N∝ℓN\propto\ell, with wave-packet rapidity widths in the window 1/ℓ≤Δθ≤11/\ell\leq\Delta\theta\leq 1. Stationary-phase and phase-linearisation analyses then show that the wave packets follow the trajectories of a classical model - the semi-classical Bethe model, point particles travelling at v(θ)v(\theta) and displaced backwards by φ(θ−α)\varphi(\theta-\alpha) whenever they overtake a particle of rapidity α\alpha (a Wigner time delay, exactly as solitons scatter in integrable PDEs) - and that the density of wave packets satisfies the GHD equation. The paper also derives an explicit formula for the action of the spectral phase-space density operator ρ^(x,θ)\hat{\rho}(x,\theta) on Bethe states (shown to generate the local conserved densities) and an exact operatorial fixed-point identity for the quasi-particle height field, a normal-ordered (positions ordered to the left of rapidities, the t-classical ordering) version of the classical self-consistent equation for the counting function N(t,x,θ)N(t,x,\theta).

The gaps are precisely identified. Each approximation - stationary phase (permutation sector by sector), Bethe-phase linearisation, the interacting Wigner-function route, and the operator route - is controlled only for fixed permutation σ\sigma, whereas expectation values contain double sums over σ,τ∈SN\sigma,\tau\in\mathcal{S}_N with fast-oscillating phases eiℓ(Φˉ(xσ,θ)−Φˉ(xτ,θ))e^{i\ell(\bar{\Phi}(\mathbf{x}_{\sigma},\boldsymbol{\theta})-\bar{\Phi}(\mathbf{x}_{\tau},\boldsymbol{\theta}))}. The working assumption that these phases force σ=τ\sigma=\tau is unjustified: there are enormously many permutations exchanging nearby particles within a range ℓ\sqrt{\ell} in rapidity space. Keeping only diagonal sectors also contradicts physics - the emergent fermionic (Pauli) statistics of the quasi-particles, which arises from the antisymmetry of ψ\psi in θ\boldsymbol{\theta} - since it would allow arbitrarily many wave packets to be packed into one region. In the Wigner-function route the same neglect visibly destroys normalisation: the image complement Σ\Sigma of the particle-position map (the forbidden region of the contracted coordinates yi=xi+12∑j≠isgn(xi−xj)φ(θi−θj)y_i=x_i+\tfrac{1}{2}\sum_{j\neq i}\mathrm{sgn}(x_i-x_j)\varphi(\theta_i-\theta_j)) is a macroscopic set whose measure is lost, and it is unknown whether the neglected eO(1)e^{\mathcal{O}(1)} corrections restore unitarity. On the operator route the fixed-point identity is exact, but deriving the GHD equation for averaged spectral densities from it requires proving that normal ordering becomes negligible at macroscopic scales and that large-scale observables factorise (classicalise) in long-wavelength states - both unproven. The authors identify the missing tool as an analog of large-deviation theory (the mathematical framework controlling exponential scalings of the probabilities of empirical measures) valid for oscillatory integrals and able to encode particle statistics.

For comparison: GHD-type equations have been derived rigorously for classical systems (hard-rod gases, the box-ball cellular automaton, and recently the Toda lattice) and for noninteracting quantum chains; for interacting quantum systems no microscopic derivation exists at any level of rigour. This is the foundational open problem on the hydrodynamic side of the field.

Problem Statement

Starting from the microscopic Bethe wave functions of an interacting quantum integrable model - the repulsive Lieb-Liniger model being the paradigm - derive, with mathematically controlled approximations, the Euler-scale GHD equation

∂tρp(t,x,θ)+∂x(veff(t,x,θ) ρp(t,x,θ))=0,\partial_{t}\rho_{\rm p}(t,x,\theta)+\partial_{x}\big(v^{\rm eff}(t,x,\theta)\,\rho_{\rm p}(t,x,\theta)\big)=0,

with

veff(θ)=v(θ)+∫dα  φ(θ−α) ρp(α) (veff(α)−veff(θ)),v^{\rm eff}(\theta)=v(\theta)+\int \mathrm{d}\alpha\;\varphi(\theta-\alpha)\,\rho_{\rm p}(\alpha)\,\big(v^{\rm eff}(\alpha)-v^{\rm eff}(\theta)\big),

for the expectation values of the conserved densities (equivalently, of the spectral phase-space density operator ρ^(x,θ)\hat{\rho}(x,\theta)) in long-wavelength states, in the scaling limit ℓ→∞\ell\to\infty, N∝ℓN\propto\ell at finite density. 'Controlled' means that every approximation step carries an error bound or a precise asymptotic statement; heuristic cancellations may not be assumed. Concretely, a solution must complete at least one of the four routes proposed by Doyon and Hübner - stationary phase, Bethe-phase linearisation and the interacting Wigner-function route (the three Schrödinger-picture routes grouped under (a) below), and the operator route (b) - or an equivalent route; completing any single one of these four routes to the stated standard suffices:

(a) Wave-function routes (Schroedinger picture), any one of which suffices: control the combined contribution of the N!N! permutation sectors in the large-scale asymptotics of superpositions of Bethe wave functions - replacing the unjustified assumption that fast-oscillating phases force σ=τ\sigma=\tau by a theorem (the sought oscillatory large-deviation framework, which must also encode the fermionic statistics of the quasi-particles) - and show that the resulting expressions satisfy exact normalisation and unitarity, accounting for the measure of the image-complement region Σ\Sigma (the region of contracted-coordinate space not attained by the particle-position map, whose measure the working approximations discard) and for the behaviour of the subleading eO(1)e^{\mathcal{O}(1)} corrections.

(b) Operator route (Heisenberg picture): prove that the exact, normal-ordered operatorial fixed-point identity for the quasi-particle height field implies the GHD equation for averaged spectral densities, by establishing that t-classical normal ordering (normal ordering with position operators ordered to the left of rapidity operators) can be dropped at macroscopic scales and that the relevant large-scale observables factorise in long-wavelength states.

The common objective of the routes is one and the same: a derivation of the GHD equation for averaged conserved densities from the microscopic dynamics with all steps controlled. The permutation-sum control, the normalisation fix, and the normal-ordering and factorisation theorems are the route-specific missing steps, not independent deliverables; completing any single route to the stated standard solves the problem. The completion boundary is the Euler-scale equation itself: extracting diffusive or genuinely quantum corrections from the microscopic construction is a natural continuation of the programme but is not required for completion.

References

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