Theory of finite-size validity of integrable hydrodynamics: why very small Toda chains reproduce thermodynamic ballistic correlation profiles

Generalized hydrodynamics (GHD) describes the ballistic, Euler-scale correlations of integrable many-body systems in the thermodynamic limit. A 2022–2023 study introduced a purely spectral finite-N route — solving the linearized Euler equations of the classical Toda chain with covariance matrices obtained by Monte Carlo sampling of the Gibbs-distributed Lax matrix — and found that already for chains with $N=8$ and $N=16$ particles the resulting ballistic peak positions fall, with high precision, on the thermodynamic GHD profile confirmed independently by large-scale molecular dynamics. The phenomenon is documented but left without conditions, mechanism, or scaling: hydrodynamics contains no small parameter that would justify agreement at $N\approx 8$, and the companion molecular-dynamics study explicitly states that a theoretical explanation seems to be missing. The open problem is to construct the finite-size theory of integrable hydrodynamics: (i) identify the mechanism and conditions of validity under which the discrete generalized velocities of small integrable chains approximate the continuum GHD prediction, and quantify the $N$-dependence of peak positions, weights and shapes, including the possible even–odd particle-number effect in the momentum correlations noted by the source; and (ii) settle whether the construction extends to long-range integrable systems of Calogero–Sutherland type, where conservation laws lose spatial locality. Thermodynamic building blocks exist — TBA-based GHD for Toda and Calogero models, central limit theorems for Lax-matrix statistics, a TBA prediction for quasiparticle diffusion in the Toda fluid — but none addresses the finite-N question.

Background

The classical Toda chain is a one-dimensional Hamiltonian lattice of particles coupled by exponential nearest-neighbour interactions, with Hamiltonian $H=\sum_i \left[p_i^2/2 + e^{-(r_{i+1}-r_i)}\right]$ in terms of momentum and stretch variables $p_i$ and $r_i$. It is integrable: its equations of motion can be cast in Lax form $\dot L=[L,M]$ for an $N\times N$ tridiagonal matrix $L$ (the Lax matrix) built from the momenta and stretches, which implies that the traces $Q_n=\mathrm{tr}\,L^n$, $n=1,\dots,N$, form an extensive family of conserved charges. In real space these charges have rapidly growing spatial support (they are increasingly non-local), although the lowest ones — momentum, energy and stretch — are local.

Because of the extensive conservation laws, transport in the Toda chain is ballistic: densities and currents of the conserved charges propagate on the Euler scale, meaning space-time positions with $x/t$ of order one. The appropriate equilibrium statistical ensemble is the generalized Gibbs ensemble (GGE), a Gibbs-like measure over all conserved charges, and generalized hydrodynamics (GHD), developed in 2016 by Castro-Alvaredo, Doyon and Yoshimura and by Bertini, Collura, De Nardis and Fagotti, is the corresponding hydrodynamic theory: in the thermodynamic limit $N\to\infty$ the dynamics of local observables obeys an infinite coupled system of conservation laws solvable through the thermodynamic Bethe ansatz (TBA), a self-consistent integral equation fixing the quasiparticle density of states and their effective velocities. Linearized GHD predicts equilibrium space-time correlation functions as superpositions of ballistic peaks: the peak velocities are the eigenvalues of the hydrodynamic projection matrix $A=C^{-1}F$, where $C$ is the covariance matrix of the conserved charges and $F$ the matrix of their current correlations with the charges; the peak weights are the Drude weights (the ballistic transport weights of charge-current combinations); and full peak shapes require the diffusion matrix $D$ and noise matrix $B$ of fluctuating hydrodynamics, which produce the sub-ballistic broadening.

Kundu (2023) introduced an independent, purely spectral route to these correlations at finite $N$. At Gibbs equilibrium the Toda Lax matrix is itself a random tridiagonal matrix with independent entries (Gaussian diagonal, chi-square distributed off-diagonal), so its charge-covariance and current-correlation matrices can be estimated by direct Monte Carlo sampling of $L$ without running the dynamics. Solving the linear Euler equations for the conserved quantities of a chain whose Lax matrix has dimension of order $N$ then yields a finite-$N$ correlation function that is a finite sum of delta peaks located at $N+1$ 'generalized velocities' — the eigenvalues of the finite hydrodynamic projection matrix built from the $N+1$ conserved quantities of the source construction (the $N$ trace charges $\mathrm{tr}\,L^n$ together with the stretch charge). Surprisingly, for $N=8$ and $N=16$ at temperature $T=1$ and pressure $P=1$ these peak positions fall, with high precision, on the thermodynamic profile obtained both from molecular dynamics with $N=2048$ particles and from TBA-based GHD. The agreement is asymmetric: peak positions are reproduced (an earlier truncation study by Kundu and Dhar had already found that retaining only the three local Gibbs charges reproduces the motion of the peaks to 5%), but peak shapes are captured poorly, with a possible even–odd particle-number effect in the momentum correlations. Kundu also noted that the approach is restricted to the Euler scale (the diffusive and noise corrections are omitted) and that its extension to long-range integrable systems such as the Calogero–Sutherland models (many-body systems of particles interacting through inverse-square pair potentials, whose interactions do not decay with distance) is unexplored.

The observation is a genuine puzzle. Hydrodynamic descriptions are derived and validated in the thermodynamic limit, and there is no small parameter at $N\approx 8$ that would justify the observed accuracy; an explanation must connect the random-matrix statistics of small tridiagonal Lax matrices to the continuum of GHD normal modes. The state of the adjacent theory is as follows. The companion molecular-dynamics study of Mazzuca, Grava, Kriecherbauer, McLaughlin, Mendl and Spohn (2023) confirmed linearized GHD at $N=3000$–$4000$ to within 3.5% with no adjustable parameters and stated, regarding Kundu's small-$N$ note that the delta peaks lie on the GHD profile with high precision, that a theoretical explanation "seems to be missing". Rigorous central limit theorems for moments of Lax matrices of integrable systems (Mazzuca and Memin, 2024) provide thermodynamic fluctuation theory, but not small-$N$ profiles. On the diffusive side, the hydrodynamic diffusion matrix of integrable systems is understood in general terms (De Nardis, Bernard and Doyon, 2018), and for the Toda fluid a TBA prediction for quasiparticle diffusion has recently been confirmed numerically (Chahal et al., 2026) — all at thermodynamic scale. For the Calogero family, thermodynamic GHD is worked out in Spohn's monograph (2024), including the rational/trigonometric Calogero–Moser models where the two-particle scattering shift (the position shift two quasiparticles imprint on each other as they cross) vanishes and locality of the conservation laws fails, and the thermodynamic Lax density of states of the short-range hyperbolic Calogero fluid is known (Kethepalli et al., 2025); the quantum Calogero–Sutherland dynamic structure factor has been computed from sum rules and Monte Carlo (Lleopart Motis et al., 2025). None of these addresses the finite-$N$ question, which for long-range models is sharpened by the loss of charge locality on which the finite-N spectral construction relies.

Problem Statement

Construct the finite-size theory of integrable hydrodynamics around one objective: explain when and why the hydrodynamic description derived in the thermodynamic limit is already quantitatively accurate for very small systems. The construction is to be established for the classical Toda chain, the paradigmatic case in which the phenomenon is documented; its extension beyond the Toda chain is the Calogero–Sutherland question of part (3). Concretely: (1) Mechanism and conditions — identify why the eigenvalues of the finite-N hydrodynamic projection matrix of the Toda chain at Gibbs equilibrium (built from the covariance and current-correlation matrices of the $N+1$ conserved quantities of the source construction — the $N$ trace charges $\mathrm{tr}\,L^n$ together with the stretch charge — sampled through the Gibbs-distributed Lax matrix) fall on the continuum of GHD normal-mode velocities already at $N=8$–$16$, and characterize the conditions of validity (thermodynamic state, retained charges, observable class) together with the scaling with $N$ of the approach of the discrete generalized velocities and their weights to the thermodynamic profiles. (2) Finite-N deviations — explain the deviations that are observed but unexplained: the peak shapes and heights, which the finite-N construction captures poorly even where the peak positions are accurate, and the possible even–odd particle-number effect in the momentum correlations, which the source could not rule out and which has not been analyzed. (3) Long-range boundary of validity — establish whether the small-N spectral construction extends to long-range integrable systems of Calogero–Sutherland type (particles interacting through inverse-square pair potentials that do not decay with distance), where the two-particle scattering shift vanishes and conservation laws lose spatial locality. A complete solution requires all three of (1)–(3); each is independently attackable, but none alone resolves the task, and the even–odd particle-number effect together with the three validity dimensions are components of (1)–(2) rather than separate optional items.

References

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