Domain of validity of typical thermalization in large harmonic systems: frequency degeneracies, eigenvector localization, matrix classification, finite-size effects, and the ETH connection

Cattaneo, Baldovin, Lucente, Muratore-Ginanneschi and Vulpiani [Phys. Rev. Research 7, L032002 (2025)] proved that thermalization of mode energies is overwhelmingly likely (typical) in large classical and quantum harmonic systems: for the energies of randomly rotated normal modes, or of local modes of random quadratic Hamiltonians, long-time averages equal microcanonical predictions with fluctuations vanishing as $N\to\infty$, with no role played by chaos or ergodicity. The theorem, however, rests on non-degenerate normal-mode frequencies, eigenvector delocalization (proved only for random-matrix ensembles and heuristically for the linear chain), bounded inverse spectral moments, and the thermodynamic limit, and it explicitly defers the connection to the eigenstate thermalization hypothesis (ETH). The open problem is to delimit the domain of validity of this chaos-free thermalization mechanism: determine whether and under what conditions typicality survives frequency degeneracies and partial eigenvector localization, classify which coupling matrices (including deterministic, physically relevant ones) guarantee the delocalization the proof needs, characterize finite-$N$ corrections to the approach-to-typicality, and make the relation to ETH-based thermalization precise for quadratic and near-integrable systems. These gaps remain open.

Background

A harmonic (quadratic) system of $N$ degrees of freedom is one whose Hamiltonian, after diagonalization into normal modes (collective oscillation patterns that decouple the dynamics), reads $H=\sum_k\omega_k o_k^*o_k$; in classical mechanics $o_k$ are complex canonical variables of the modes, in quantum mechanics bosonic or fermionic ladder operators, and each mode evolves as $o_k(t)=e^{-i\omega_k t}o_k(0)$. Such systems are integrable — the mode energies $E_k=\omega_k o_k^*o_k$ are all conserved — so they lie outside chaos- and ergodicity-based explanations of thermalization, and their equilibrium is normally described by a generalized Gibbs ensemble (GGE, the ensemble fixing all conserved mode energies) rather than by the microcanonical ensemble (the uniform distribution on the constant-total-energy shell). A century-old alternative, associated with Khinchin and extended by Mazur and van der Linden, holds that statistical mechanics needs only a large number of degrees of freedom and observables that are sums over many of them: for such observables, the overwhelming majority of microcanonical initial conditions already take the ensemble-average value. Numerical studies of the harmonic chain [Cocciaglia et al., Physica A 601 (2022); Baldovin et al., Physica A 630 (2023)] gave evidence in this spirit, and related typicality results for macroscopic quantum systems were proved by Teufel, Tumulka and Vogel [J. Stat. Phys. 190 (2023)].

Cattaneo et al. [Phys. Rev. Research 7, L032002 (2025)] proved a dynamical version for harmonic systems. An observable (dynamical variable) is said to thermalize if its long-time average $\overline{O}_\infty$ equals its microcanonical ensemble average $\langle O\rangle$ for almost all initial conditions, with fluctuations that vanish in the large-$N$ limit; for the mode energies, with $O_k$ the long-time average of the mode-$k$ energy and $\tilde E_k$ the mode-$k$ energy itself, the relative fluctuation is $\epsilon_k=[\langle(O_k)^2\rangle-\langle\tilde E_k\rangle^2]/\langle\tilde E_k\rangle^2$. The paper shows that for the energies of rotated modes $\tilde o_j=\sum_k C_{jk}o_k$ ($C$ an orthogonal or unitary matrix) and for the local particle energies of a quadratic Hamiltonian $H=\sum_{jk}M_{jk}\tilde o_j^*\tilde o_k$, the disorder average of $\epsilon_k$ tends to zero and its relative variance vanishes, $\mathbb{E}[\epsilon_k]\to0$ and $\mathrm{Var}[\epsilon_k]/\mathbb{E}[\epsilon_k]^2\to0$ for $N\gg1$ (with $\mathbb{E}$ and $\mathrm{Var}$ taken over realizations of the random rotation or matrix), so thermalization is typical when initial conditions are drawn from the microcanonical distribution, and even when only a subset $1\ll N^*\ll N$ of modes is initially excited. Roughly equal initial mode populations instead yield equipartition without thermalization. In quantum mechanics the same conclusions hold for fermions and hard-core bosons (with vanishing time fluctuations of microscopic observables), making the mechanism a bridge between classical and quantum thermalization.

The theorem rests on four hypotheses. (1) Non-degenerate normal-mode frequencies ($\omega_j\neq\omega_k$ for $j\neq k$), a standard assumption in quantum thermalization theory; it ensures that in the long-time average all cross-terms oscillating at nonzero frequency differences $\omega_j-\omega_k$ dephase, i.e. average to zero, so that no resonant contribution survives. (2) Eigenvector delocalization: the rows of the diagonalizing matrix $C$ are effectively uniformly distributed over the unit sphere — their components are all of comparable size $1/\sqrt{N}$ rather than concentrated on a few entries, a property quantified by the inverse participation ratio $\mathrm{IPR}_j=\sum_k C_{jk}^4$ being of order $1/N$ for delocalized vectors versus $O(1)$ for localized ones. Delocalization is proved for random-matrix ensembles (GOE/GUE, the Gaussian orthogonal and unitary ensembles; Wigner matrices with independent entries; the Rosenzweig-Porter model, a random-matrix family with tunable interpolation between nearly diagonal and fully random matrices) and heuristically for the deterministic linear chain (via equidistribution of the sine-squared eigenvector components); for broader classes it is only conjectured [Rudelson, arXiv:1707.08461], while for Anderson-localized eigenvectors (eigenvectors confined to a small region near one site by disorder, as in Anderson's lattice model) delocalization fails and the typicality argument does not go through. (3) Bounded inverse spectral moments, $\lim_{N\to\infty}\frac{1}{N}\sum_k\omega_k^{-n}<\infty$ for $n=1,2,4$ (or an infrared cutoff $\omega_k\geq\omega_m$). (4) The thermodynamic limit $N\to\infty$; only leading $O(1/N)$ asymptotics are derived.

The standard quantum route to thermalization is instead the eigenstate thermalization hypothesis (ETH): individual energy eigenstates already carry thermal values of observables. For quadratic Hamiltonians this line has been developed for random and quantum-chaotic single-particle spectra [Łydżba et al., PRL 125 (2020); Łydżba et al., PRL 131 (2023)], where generalized (GGE-type) thermalization is generally needed, and weak eigenstate thermalization was found in a fully integrable open system [Ptaszyński and Esposito, PRE 111 (2025)]. The source paper's own conclusion poses the connection between its typicality mechanism and ETH-based reasoning in integrable systems as an open direction.

Problem Statement

Delimit the domain of validity of typical thermalization in large harmonic systems — replacing each hypothesis of the typicality theorem of Cattaneo et al. by either a proven generalization or a sharp failure boundary. Concretely:

1. Degeneracies and partial localization. Determine whether and how the typicality of thermalization of rotated-mode and local-mode energies is modified when normal-mode frequencies are degenerate (so that resonant denominators $\omega_k-\omega_{k'}$ vanish in the dephasing sums) or when the eigenvectors are partially localized (inverse participation ratio between the delocalized $\sim1/N$ and localized $O(1)$ regimes): give conditions on the degeneracy pattern and localization strength under which typical thermalization still holds, or construct explicit systems and initial-condition ensembles with partially degenerate frequencies or partially localized eigenvectors for which it fails; the fully degenerate spectrum alone does not settle this facet, and both the degeneracy and the partial-localization sub-cases must be addressed.
2. Matrix classification. Identify which classes of coupling matrices $M$ — beyond the random ensembles already covered and the linear chain treated heuristically — guarantee the eigenvector delocalization that the typicality proof requires; in particular, delimit for which physically relevant deterministic matrices delocalization holds, fails, or is borderline, so that the delocalization hypothesis is replaced by a checkable structural criterion.
3. ETH connection. Make precise how the dynamical-typicality mechanism relates to eigenstate-thermalization-based thermalization: state and prove (or refute) conditions under which the typicality statement implies, is implied by, or is independent of ETH-type statements for quadratic and near-integrable quantum systems, unifying the chaos-free classical mechanism with the eigenstate-based quantum framework.
4. Finite-size corrections. Characterize the finite-$N$ corrections to the typicality statements: explicit bounds or scaling laws for the approach-to-typicality — the relative fluctuations $\epsilon_k(N)$ and the normalized effective number of thermally active modes $n_{\mathrm{eff}}(N)=\exp(-\sum_k u_k\log u_k)/N$, where $u_k$ is the fraction of the total time-averaged energy carried by mode $k$ and $n_{\mathrm{eff}}=1$ at exact equipartition — at finite, experimentally relevant system sizes, beyond the leading $O(1/N)$ asymptotics and covering degenerate spectra and cutoff frequency distributions.

Each of the four sub-questions above is independently resolvable: a proven generalization or a sharp failure boundary in any one of them settles that facet of the domain of validity — frequency spectra, eigenvector localization structures, matrix classes, or system sizes — while the other facets remain open, so the full delimitation of where majority relaxation holds in harmonic systems without chaos or mixing requires addressing all four.

References

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