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Ergodicity and hydrodynamics: from quantum to classical spin systems

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Pith's one-line read For ergodic classical spin systems, the late-time autocorrelation of any local observable is determined by energy transport and the observable's thermodynamic energy-overlap order.

desk verdict Credible classical extension of the relaxation-overlap inequality with clean numerics, but the headline equality is not derived and the 'generic observable' claim rests on an untested hydrodynamic projection assumption. read the letter →

arxiv 2509.04098 v1 pith:PYQRURIV submitted 2025-09-04 cond-mat.stat-mech cond-mat.quant-gasnlin.CDquant-ph

Jiaozi Wang , Luca Capizzi , Dario Poletti , Leonardo Mazza This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.quant-gasnlin.CDquant-ph MSC 82C0582C7082B20 PACS 05.20.-y05.60.-k75.10.Hk
keywords classicalspinsystemsergodicityhydrodynamictailsrelaxation-overlapinequalityautocorrelationfunctionsdynamicalcriticalexponentenergytransportlong-rangeIsingmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that in classical spin systems with ergodic dynamics and energy as the only conserved charge, the late-time decay of any local observable's autocorrelation function is fixed by two numbers: the dynamical critical exponent z of energy transport and the lowest order m at which the observable's thermal expectation depends on energy density. Concretely, the hydrodynamic tail decays as t^{-ν} with ν = dm/z, and the long-time finite-size plateau obeys C(∞) ~ L^{-m}. The authors verify this in one- and two-dimensional Ising-type spin models, including long-range models with anomalous transport, and show that the same relation also fixes the system-size saturation. The classical derivation replaces the eigenstate thermalization hypothesis used in the quantum case by ordinary ergodicity, so the relaxation-overlap logic applies across quantum and classical many-body physics. If the relation holds, the late-time dynamics of a generic observable can be predicted from equilibrium thermodynamics alone.

What carries the argument

Two mechanisms carry the argument. First, ergodicity in the form that long-time averages of any observable equal its microcanonical expectation at the initial energy: applied to the autocorrelator, this turns its late-time plateau into the variance of O across the thermal ensemble, which scales as L^{-m} when O(ε) ~ ε^m. Second, hydrodynamic projection: a generic local observable is dominated in its slow dynamics by its overlap with the energy density and its powers, whose spreading follows the scaling ⟨h(x,t)h(0,0)⟩ ~ t^{-1/z}F(x/t^{1/z}). Since energy spreads over a region of size t^{1/z}, finite-size saturation occurs at t ~ V^{z/d}; monotone decay then forces ν ≤ dm/z, and the numerical

What would settle it

In a one-dimensional short-range spin chain with demonstrably diffusive energy transport (z = 2) and no other conserved charge, measure the autocorrelation of S^x_j, whose energy-overlap order is m = 1. The prediction is an algebraic tail C(t) ~ t^{-1/2} followed by a plateau ~ 1/L. A clean t^{-ν} tail with ν differing from 1/2 by more than the numerical uncertainty, or a plateau that fails to scale as 1/L, would falsify the formula.

Watch

Extended reading notes

Core claim

On the paper's own terms: for a chaotic, ergodic classical spin system at high temperature with energy as the only conserved charge, define m such that the thermal expectation O(ε) grows as ε^m near the relevant energy density. Then the equilibrium autocorrelator exhibits a hydrodynamic tail ⟨O(t)O⟩_c ~ t^{-dm/z}, where z is the dynamical critical exponent of energy spreading, and its infinite-time plateau decays as L^{-m}. The plateau scaling is derived from ergodicity through a saddle-point expansion of the microcanonical variance; the connection between plateau and tail follows from hydrodynamic projection and monotone decay of the autocorrelator, yielding the relaxation-overlap inequalit

Load-bearing premise

The result rests on the hydrodynamic projection assumption: at late times a generic local observable's autocorrelation is governed solely by its projection onto the energy density and its powers, with non-conserved modes contributing only exponentially small corrections; if another conserved quantity or a slow non-hydrodynamic mode dominates the observable, ν = dm/z can fail.

Editorial extensions

If this is right

  • The hydrodynamic tail of a local observable is not universal: in a fixed diffusive system, observables with energy-overlap orders m = 1, 2, 3, 4 decay as t^{-d/z}, t^{-2d/z}, t^{-3d/z}, t^{-4d/z} respectively.
  • The finite-size plateau of the autocorrelator scales as L^{-m}, giving a direct dynamical readout of the observable's thermodynamic overlap order.
  • In the long-range model with α = 1.5, the theory predicts an intermediate superdiffusive tail with z = 4/3 followed by a late-time diffusive tail with z = 2; the large-system numerics confirm this two-stage decay.
  • In the long-range model with α = 1.1, the dynamical exponent z = 2α - 1 persists without a crossover to diffusion, so the tail exponent is m/(2α - 1) throughout.
  • In two-dimensional diffusive systems, the same observables show tails t^{-m}, matching ν = dm/z with d = 2, z = 2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same logic should apply charge by charge: in systems with additional conserved quantities, the natural replacement is the observable's overlap with that charge's density and its powers, yielding a family of ν = dm/z relations indexed by each conservation law.
  • The formula is the saturation of an inequality; near-integrable, scarred, or slowly relaxing regimes could show ν strictly smaller than dm/z, offering an independent test of the hydrodynamic-projection assumption.
  • Because the plateau amplitude at fixed L is controlled by equilibrium thermodynamic derivatives, C(∞) scaling could be used as a diagnostic: measure the plateau at a few sizes, extract m, and compare it with the first nonvanishing energy derivative of the observable's thermal expectation.
  • For the α = 1.5 long-range model, the crossover between the z = 4/3 and z = 2 tails should occur on a timescale set by L^{z/d}; this is a quantitative prediction one could verify with time-resolved data at larger sizes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies classical spin systems with a single conserved quantity (energy) and proposes that the late-time decay exponent ν of a local observable's autocorrelator is fixed by ν = dm/z, where d is the spatial dimension, z is the energy dynamical critical exponent, and m is the order of the first nonvanishing term in the thermal expectation O(ε) around the relevant energy density. It reports extensive numerical simulations for a 1D tilted-field Ising model, a 1D long-range Ising model (α = 1.1 and 1.5), and a 2D transverse Ising model, using observables S^x_j, S^x_jS^x_{j+1}, S^x_jS^x_{j+1}S^x_{j+2}, and S^x_jS^x_{j+1}S^x_{j+2}S^x_{j+3}, with overlap orders m = 1,...,4. The finite-size plateau of the autocorrelator is shown to scale as L^{-m}, and the hydrodynamic-tail exponents are reported to be compatible with ν = dm/z. A theoretical derivation from ergodicity gives the plateau scaling and an inequality ν ≤ dm/z under additional assumptions of hydrodynamic projection and monotone decay; the equality is then inferred from the numerical fits.

Significance. If the central equality could be established, the paper would provide a clean classical analogue of the quantum relaxation-overlap relation, connecting thermodynamics, ergodicity, and hydrodynamics. The plateau scaling result (Eq. 14) is derived from ergodicity and is well supported by the numerics, and the analytical computation of the overlap order in Appendix B is a useful concrete ingredient. The numerical evidence is also broad: large systems (up to L = 200), two spatial dimensions, and several transport regimes. However, the headline relation is not actually proven in the text; the derivation yields only an inequality, and the key hydrodynamic-projection assumption is not tested independently. The manuscript therefore overstates the degree to which Eq. (13) is established for generic observables.

major comments (3)
  1. [Sec. III C, Eq. (13); Sec. IV C] Eq. (13) is stated as an equality, ν = dm/z, and the abstract claims that the late-time tail is 'determined' by z and m. But the derivation in Sec. IV C establishes only the inequality ν ≤ dm/z, and it does so under two additional assumptions (hydrodynamic projection and monotonic decay). The text itself calls this the 'relaxation-overlap inequality'. The equality is inferred from fits, not derived. Please either provide a saturation argument or reframe the central claim as an inequality plus numerical evidence of saturation for the observables studied.
  2. [Sec. IV C, hydrodynamic projection] The hydrodynamic-projection assumption is load-bearing and is not tested. For a zero-overlap observable such as O = S^y_j in the model of Eq. (10), the Hamiltonian contains no S^y, so ⟨H^m S^y_j⟩_{β=0} = 0 for all m and O(ε) = 0 identically. The theory then predicts no algebraic tail and no L^{-m} plateau, only exponentially decaying non-hydrodynamic contributions. The manuscript computes only observables with nonzero overlap m. A direct numerical test of S^y_j (or a similar symmetry-odd observable) would be needed to support the claim that Eq. (13) applies to generic local observables.
  3. [Sec. III A and III C, α = 1.5 case] The paper states that for α ≥ 1.5 energy transport is diffusive (z = 2), yet the confirmation of Eq. (13) for the α = 1.5 long-range model uses an intermediate-time exponent z = 4/3 taken from the quantum work Ref. [24], not from the energy-density scaling of this classical model. The same data also show a later tail with z = 2. It is unclear whether Eq. (13) is being tested against the asymptotic dynamical critical exponent or fit to a crossover regime with an effective exponent. The status of z = 4/3 for this classical model should be clarified, ideally by an independent measurement of the energy autocorrelator scaling.
minor comments (4)
  1. [Sec. IV C] The monotonicity assumption is introduced as a physically motivated hypothesis, but the raw autocorrelators in Figs. 2, 4, 6, and 8 show transient oscillations and clear plateau effects. The derivation of ν ≤ dm/z would benefit from a precise statement of how the averaging that restores monotonicity is performed.
  2. [Sec. III C / Fig. 9] The notation C(∞) for the long-time plateau in finite systems may be confused with the true t → ∞ limit in the thermodynamic limit. Consider using C_plateau(L) or explicitly defining it as the long-time average at fixed L.
  3. [Sec. III B] The numerical section reports the integration time step but not the total integration time or the time at which the plateau is measured for each L. This information would make the fitting procedure for ν and the plateau extraction more reproducible.
  4. [Sec. II B / figure captions] There are minor typos: 'coordin dates' should be 'coordinates', and several figure captions read 'Dashed line indicate' instead of 'Dashed lines indicate'.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: Eq. (14) is derived from ergodicity, m is analytic, and Eq. (13) is tested against numerically fitted exponents with z from external and independently re-tested sources.

full rationale

The central derivation is not circular. The plateau scaling Eq. (14), C(∞) ∼ L^{-m}, is derived in Sec. IV B from ergodicity: the late-time averaged autocorrelator is expressed as the variance of the time-averaged observable, which is then evaluated via the microcanonical/energy-shell representation (Eqs. (22)–(25)); a saddle-point expansion of O(ε) ∼ ε^m and the central-limit scaling of energy fluctuations yield ⟨(ε−ε(β))^{2m}⟩ ∼ V^{-m} (Eq. (27)), giving Eq. (28). The overlap order m is not fitted to the autocorrelator; it is computed analytically in Appendix B as the first m with ⟨H^m O⟩_{β=0} ≠ 0. The dynamical exponent z is taken from external literature for the short-range models (z=2) and for the long-range model α=1.1 (z=2α−1 from Ref. [39]); for the α=1.5 intermediate regime, z=4/3 is borrowed from the authors' prior Ref. [24], but this is independently re-tested on classical data in Fig. 4 and is therefore evidence, not a circular input. The theory section actually derives the inequality ν ≤ dm/z (Sec. IV C), and the equality ν = dm/z is an empirical saturation inferred from the numerical fits; this is a legitimate (if slightly overstated in the wording) combination of a bound with observations, not a construction. The hydrodynamic projection assumption (non-conserved modes decay exponentially) is a physical input that is not microscopically derived and is not tested on observables with zero energy overlap (e.g., S^y_j symmetric-odd operators); this is a genuine correctness/robustness risk, but it is not a circular step because no quantity in the derivation is defined in terms of the claimed result. The self-citation to Ref. [24] is substantial but load-bearing only as a source of the framework and one intermediate z value, both independently tested here; it does not reduce the central claim to the citation.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The derivation relies on standard statistical-mechanics assumptions (ergodicity, CLT, hydrodynamic scaling) plus the physically motivated hydrodynamic projection and monotonicity assumptions. The only parameter introduced ad hoc is the intermediate z=4/3 for the α=1.5 model, which is not derived for the classical system.

free parameters (1)
  • Intermediate-time effective exponent z=4/3 for the α=1.5 long-range model = 4/3
    Used to match the intermediate-time algebraic tails in Fig. 4; this value is imported from the authors' prior quantum paper [24] and is not the classical model's energy transport exponent, which is diffusive (z=2).
assumptions (6)
  • domain assumption Ergodicity / shell ergodicity: long-time averages of observables converge to microcanonical averages at fixed energy (Eq. 16)
    The entire derivation of the plateau scaling in Sec. IV B rests on this assumption; it is stated in Sec. IV A and not proven for the studied models.
  • domain assumption Energy is the only conserved quantity for the models studied
    Stated in Sec. IV A: 'we assume that no additional conserved quantities, independent from H, are present; to the best of our knowledge, this is the case for the three models studied in Sec. III.'
  • domain assumption Hydrodynamic scaling of the energy density autocorrelation (Eq. 29)
    Assumed to derive the crossover time t* ~ V^{z/d}; this is standard for diffusive or anomalous energy transport.
  • domain assumption Hydrodynamic projection: generic observables couple to energy density and its powers, with exponentially decaying non-conserved modes
    Introduced in Sec. IV C as the 'key mechanism' bridging energy transport to arbitrary observables; no microscopic derivation is given.
  • ad hoc to paper Monotonic decay of the autocorrelation function after transient oscillations
    Explicitly assumed in Sec. IV C as a 'physically-motivated hypothesis' to convert the plateau estimate into the inequality ν ≤ md/z.
  • standard math Central limit theorem for energy fluctuations in the thermal state
    Used in Eq. (27) to estimate ⟨(H/V−ε)^m⟩ scaling; standard for sums of weakly correlated variables given the clustering property.

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Cite this review

Pith. "Pith review of Ergodicity and hydrodynamics: from quantum to classical spin systems." pith.science (2026). https://pith.science/paper/PYQRURIV

@misc{pith2026250904098,
  author       = {Pith},
  title        = {Pith review of: Ergodicity and hydrodynamics: from quantum to classical spin systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PYQRURIV}},
  note         = {Machine review of arXiv:2509.04098}
}
read the original abstract

We show that in classical spin systems the precise nature of the late-time hydrodynamic tails of the autocorrelation functions of a generic observable is determined by (i) the dynamical critical exponent and (ii) the equilibrium thermodynamic properties of the corresponding observable. We provide numerical results for one- and two-dimensional systems and present theoretical considerations that only rely on the notion of ergodicity. Our result extends to the classical framework the relaxation-overlap inequality, first introduced in Capizzi et al. Phys. Rev. X 15, 011059 (2025)] for quantum many-body systems satisfying the eigenstate thermalization hypothesis.

Figures

Figures reproduced from arXiv: 2509.04098 by the authors.

Figure 1
Figure 1. ⟨O⟩β versus ⟨H⟩β/L for the observables (a) S x j ; (b) S x j S x j+1; (c) S x j S x j+1S x j+2 and (d) S x j S x j+1S x j+2S x j+3 in the one-dimensional Ising model with tilted field. The dashed line indicates the analytical prediction ε m derived in Appendix B. 101 103 10−2 10−1 C(t) L = 10, 12, . . . , 20, 200 (a) ∝ t−0.5 101 103 10−4 10−3 10−2 (b) ∝ t−1 101 103 t 10−6 10−4 10−2 C(t) (c) ∝ t−1.5 101 103 t 10−9 10… view at source ↗
Figure 2
Figure 2. C(t) versus t for the infinite temperature β = 0 state for the four observables (a) S x j ; (b) S x j S x j+1; (c) S x j S x j+1S x j+2 and (d) S x j S x j+1S x j+2S x j+3 in the one-dimensional Ising model with tilted field. Dashed line indicate ∝ t dm/z for m = 1, 2, 3, 4, respectively with z = 2 and d = 1. with a time step of δt = 0.02 (δt = 0.01 for tilted field Ising model). The initial spin configurations are … view at source ↗
Figure 3
Figure 3. ⟨O⟩β versus ⟨H⟩β/L for observables (a) S x j ; (b) S x j S x j+1; (c) S x j S x j+1S x j+2 and (d) S x j S x j+1S x j+2S x j+3 in the one￾dimensional long range (α = 1.5) Ising model with tilted field. The dashed line indicates the analytical prediction ε m derived in Appendix B. 100 102 10−3 10−2 10−1 C(t) L = 10, 12, . . . , 20, 50, 200 (a) ∝ t−0.75 ∝ t−0.5 100 102 10−4 10−2 (b) ∝ t−1.5 ∝ t−1.0 100 102 t 10−6 10−4… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: C(t) versus t for infinite temperature β = 0 for observables (a) S x j ; (b) S x j S x j+1; (c) S x j S x j+1S x j+2 and (d) S x j S x j+1S x j+2S x j+3 in one-dimensional long-range Ising model (α = 1.5). The black dashed line indicate ∝ t dm/z for m = 1, 2, 3, 4, res…
Figure 5
Figure 5. Figure 5: ⟨O⟩β versus ⟨H⟩β/L for observables (a) S x j ; (b) S x j S x j+1; (c) S x j S x j+1S x j+2 and (d) S x j S x j+1S x j+2S x j+3 in the one￾dimensional long range (α = 1.1) Ising model with tilted field. The dashed line indicates the analytical prediction ε m derived in …
Figure 8
Figure 8. Figure 8: C(t) versus t for infinite temperature β = 0 for observables (a) S x j ; (b) S x j S x j+1; (c) S x j S x j+1S x j+2 and (d) S x j S x j+1S x j+2S x j+3 in the two-dimensional transverse Ising model (ℓ × ℓ square lattice). Here L = ℓ 2 , which indicates the total numbe…
Figure 9
Figure 9. Figure 9: Long time average C(∞) versus L for (a) one￾dimensional mixed field Ising model; (b) two-dimensional transverse Ising model; (c) one-dimensional long range trans￾verse Ising model α = 1.5 and (d) one-dimensional long range transverse Ising model α = 1.1. The dashed lin…

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Reviewed August 5, 2026 · model on record in the stance chip above.