Doyon, Hübner and Yoshimura (arXiv:2312.14855, published in Annales Henri Poincaré 2026) constructed a family of classical integrable many-body systems — 'semiclassical Bethe systems' obtained as generalised TTˉT\bar{T}-deformations of free particles — whose two-body scattering shift φ(θ−θ′)\varphi(\theta-\theta') can be any continuously differentiable, non-negative, even function, and proved rigorously that their equilibrium thermodynamics is given by the thermodynamic Bethe ansatz, under the assumptions of their TBA theorem (finite total shift φtot=∫Rdθ φ(θ)\varphi_{\rm tot}=\int_{\mathbb{R}}d\theta\,\varphi(\theta), integrable e−β(x,θ)e^{-\beta(x,\theta)}, and the smallness condition e−β−φtot<π(1+4e−1−1)e^{-\beta_-}\varphi_{\rm tot}<\pi(\sqrt{1+4e^{-1}}-1)). The dynamical counterpart remains unproven: the authors argue non-rigorously that the evolution from slowly varying generalised Gibbs ensembles is governed by generalised hydrodynamics (GHD), including diffusive and dispersive corrections and correlation functions, and they identify the rigorous proof of this emergence — achieved so far only for hard rods, whose shift is constant — as an open problem, in both the 2023 preprint and the March 2026 published version. Rigorous hydrodynamic limits remain confined to hard rods and rods of random length: to date no rigorous hydrodynamic limit is known for the admissible class of shifts considered here, and the post-2023 treatments of general scattering shifts (ballistic macroscopic fluctuation theory, exact microscopic diffusive theories, wave-packet simulations) are formal or explicitly non-rigorous. Proving the hydrodynamic limit for arbitrary admissible shifts, and settling the rigorous form of the subleading corrections and correlations, is therefore open.

Background

One-dimensional integrable many-body systems are finely tuned dynamics constrained by infinitely many conservation laws. Two structural consequences are central here: many-body scattering is elastic and factorised, so that the total scattering shift accumulated by each particle is a sum of two-body shifts φ(θ−θ′)\varphi(\theta-\theta') depending only on the asymptotic momenta θ,θ′\theta,\theta'; and equilibrium states are generalised Gibbs ensembles (GGEs), Gibbs-like measures built on the whole family of conserved quantities. A TTˉT\bar{T}-deformation deforms a Hamiltonian by a bilinear product of conserved currents and preserves integrability; its generalisation proposed by Doyon, Durnin and Yoshimura (SciPost Physics 13, 072 (2022)) — using the full space of pseudolocal (quasi-local, spatially screened) currents — was conjectured to realise integrable systems with essentially arbitrary two-body scattering shifts, but the locality of the resulting models was unclear. Doyon, Hübner and Yoshimura resolved this at the classical level. They start from NN free Galilean particles and apply the canonical transformation generated by Φ(x,θ)=∑ixiθi+12∑i≠jψ(xi−xj,θi−θj),\Phi(\boldsymbol{x},\boldsymbol{\theta})=\sum_i x_i\theta_i+\frac{1}{2}\sum_{i\neq j}\psi(x_i-x_j,\theta_i-\theta_j), where the freely chosen deformation function ψ\psi obeys ψxθ(x,θ)≥0\psi_{x\theta}(x,\theta)\geq 0, ψ(−x,−θ)=ψ(x,θ)\psi(-x,-\theta)=\psi(x,\theta) and ψx(x,θ)=0\psi_x(x,\theta)=0 for ∣x∣≥ε|x|\geq\varepsilon (finite interaction range ε\varepsilon). The dynamics is defined by free motion of the transformed coordinates, yi(t)=yi+θity_i(t)=y_i+\theta_i t: solving yi(t)=xi+∑j≠iψθ(xi−xj,θi−θj)y_i(t)=x_i+\sum_{j\neq i}\psi_\theta(x_i-x_j,\theta_i-\theta_j) for x(t)\boldsymbol{x}(t) gives the interacting trajectories — a 'tracer dynamics' in which each particle carries its asymptotic momentum forever. They prove rigorously that these are Liouville integrable Hamiltonian systems with finite-range interactions, with elastic factorised scattering whose two-body shift φ(θ)=∫Rdx ψxθ(x,θ)\varphi(\theta)=\int_{\mathbb{R}}dx\,\psi_{x\theta}(x,\theta) can be any continuously differentiable, non-negative, even function, and that their GGEs are described by the thermodynamic Bethe ansatz (TBA) with Maxwell–Boltzmann statistics; the TBA theorem is established under the additional assumptions φtot=∫Rdθ φ(θ)<∞\varphi_{\rm tot}=\int_{\mathbb{R}}d\theta\,\varphi(\theta)<\infty, ∫R2dx dθ e−β(x,θ)<∞\int_{\mathbb{R}^2}dx\,d\theta\,e^{-\beta(x,\theta)}<\infty and e−β−φtot<π(1+4e−1−1)e^{-\beta_-}\varphi_{\rm tot}<\pi(\sqrt{1+4e^{-1}}-1) (Theorem 2.3 of arXiv:2312.14855), so constant shifts and large-shift or low-temperature GGEs lie outside the proven equilibrium regime. Hard rods — rods of fixed length aa exchanging velocities at contact, the classic integrable gas of one-dimensional statistical mechanics — carry the constant shift φ=−a\varphi=-a in this sign convention, a rod removing space available to the others, whereas the present family has φ≥0\varphi\geq 0, a positive shift adding effective space; its constant-shift member is thus the sign-flipped hard-rod gas of 'negative rod length', and the general models replace this constant effective rod length by a momentum-dependent one. Generalised hydrodynamics (GHD) is the Euler-scale hydrodynamic theory of integrable systems (the Euler scale being the ballistic regime x∼tx\sim t on which each fluid cell remains in local equilibrium): with ρ(x,θ)\rho(x,\theta) the local quasi-particle density, GHD predicts ∂tρ+∂x(veff[ρ] ρ)=0\partial_t\rho+\partial_x(v^{\rm eff}[\rho]\,\rho)=0 with effective velocity fixed by the dressing equation θ=veff(θ)−∫dθ′ φ(θ−θ′) ρ(θ′)(veff(θ)−veff(θ′))\theta=v^{\rm eff}(\theta)-\int d\theta'\,\varphi(\theta-\theta')\,\rho(\theta')(v^{\rm eff}(\theta)-v^{\rm eff}(\theta')) (up to the 2π2\pi normalisation of φ\varphi fixed by the TBA conventions of the source paper). The only fully rigorous derivation of GHD from microscopic dynamics for an interacting particle system remains the hard-rod hydrodynamic limit of Boldrighini, Dobrushin and Suhov (1983), sharpened through the Lévy–Chentsov-field picture of Ferrari, Franceschini, Grevino and Spohn (the Lévy–Chentsov field being the Gaussian random field describing the displacement fluctuations of the rods) and the diffusive-fluctuation and multitime results of Ferrari and Olla for rods of random length. The source paper shows that its systems reproduce GHD only non-rigorously — via the explicit x→yx\to y map, which trivialises the dynamics in the same way the hard-rod contraction map does — and states that extending the rigorous proof beyond hard rods, to the wide family of admissible scattering shifts, is open. The question is sharpened by recent exact-but-non-rigorous hard-rod calculations showing that the diffusive corrections to GHD take an unconventional form: coupled, time-reversible equations for one- and two-point functions rather than a Navier–Stokes-type gradient expansion, so the subleading structure a proof must capture is itself partly unexpected.

Problem Statement

For the family of semiclassical Bethe systems constructed by Doyon, Hübner and Yoshimura — canonical transformations of free Galilean particles generated by Φ(x,θ)=∑ixiθi+12∑i≠jψ(xi−xj,θi−θj)\Phi(\boldsymbol{x},\boldsymbol{\theta})=\sum_i x_i\theta_i+\frac{1}{2}\sum_{i\neq j}\psi(x_i-x_j,\theta_i-\theta_j) with ψxθ≥0\psi_{x\theta}\geq 0, ψ\psi even, and finite interaction range, giving two-body scattering shift φ(θ)=∫dx ψxθ(x,θ)\varphi(\theta)=\int dx\,\psi_{x\theta}(x,\theta) — prove rigorously the emergence of generalised hydrodynamics. The core requirement is the Euler-scale hydrodynamic limit: for initial conditions distributed according to slowly varying GGEs (GGE Lagrange parameters β(x/L,θ)\beta(x/L,\theta) varying on the large scale LL, in the notation of the source papers), show that in the scaling x∼t∼Lx\sim t\sim L the local expectations of the conserved densities converge, with controlled errors, to the solution of the GHD equation with TBA equation of state and effective velocity dressed by φ\varphi, for the whole admissible class of shifts φ\varphi (continuously differentiable, non-negative, even), including both the constant-shift case and the non-constant shifts. Within the same rigorous program, establish the subleading structure that the source explicitly asks these models to reproduce as part of GHD: the diffusive-order corrections and the Euler-scale correlation functions — including whether they take the coupled, time-reversible, non-Navier–Stokes form found exactly for hard rods — and, where feasible, the dispersive order. A resolution of the task therefore requires both the Euler-scale limit for every admissible shift and this subleading structure — the diffusive-order corrections and the Euler-scale correlation functions; the dispersive order is an optional extension. The single overarching objective is to decide, rigorously, whether the hydrodynamic limit exists throughout the admissible class of shifts and coincides with GHD in all its features. A rigorous negative result — an admissible φ\varphi for which the limit fails or differs from GHD — would decide the question in the negative; heuristic or purely numerical derivations do not decide it.

References

  1. Benjamin Doyon, Friedrich Hübner, Takato Yoshimura, 'Generalised TTˉT\bar{T}-deformations of classical free particles', arXiv:2312.14855; published in Annales Henri Poincaré (2026), DOI 10.1007/s00023-026-01684-1, https://arxiv.org/abs/2312.14855
  2. Benjamin Doyon, Joseph Durnin, Takato Yoshimura, 'The Space of Integrable Systems from Generalised TTˉT\bar{T}-Deformations', SciPost Physics 13, 072 (2022), DOI 10.21468/SciPostPhys.13.3.072, https://doi.org/10.21468/SciPostPhys.13.3.072
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