{"id":"f2c49ffd-bde2-46ba-afde-459c55db8af7","arxiv_id":"2506.03353","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Large-scale hydroelastic waves driven by small-scale random forcing show Rayleigh-Jeans equilibrium spectra over about a decade, with zero net energy flux and measurable effective temperature, entropy, and heat capacity.","lead":"An experiment on waves in an elastic sheet floating on water shows that the large-scale motions reach a statistical equilibrium, even while small scales are driven randomly. The observed spectra match the Rayleigh-Jeans prediction from wave turbulence theory, and the system can be assigned an effective temperature, entropy, and heat capacity.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The equilibrium attribution hinges on the absence of direct wavemaker leakage into the large-scale band; the paper reports no drive-consistent low-frequency coherence or accelerometer spectrum, so a transfer-function test should precede full acceptance.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: direct forcing leakage or residual tank eigenmode contamination could mimic the equilibrium signatures. I agree, and I would sharpen the requirement into a quantitative transfer-function and coherence check rather than relying on the reported zero-flux estimate, which is indirect and lacks error bars. Other potential issues, such as the partly circular temperature estimates and the gravity contribution to the low-frequency dispersion, are secondary: they would alter the interpretation or the fitted coefficients, but they do not attack the core attribution as directly as a possible low-frequency forcing tail. The paper does have real independent support: the spatiotemporal spectrum concentrates around the linear dispersion relation, Gaussian statistics are observed, the frequency and wavenumber spectra collapse over multiple tensions, and a zero net flux is measured. These make the equilibrium claim plausible, but they do not by themselves exclude a weak direct drive in the equilibrium band. The reader's CONDITIONAL verdict already captures this uncertainty; the missing measurement is precisely the condition that should be satisfied before full acceptance. Therefore the verdict should remain unchanged, with the added recommendation that the authors report the wavemaker acceleration spectrum and drive-noise coherence explicitly.","tokens_in":8857,"tokens_out":14262,"duration_ms":183871,"concrete_test":"Under the reported forcing conditions, record the wavemaker accelerometer signal a(t) and the LDV elevation η(t) simultaneously for at least 5 minutes. Compute the transfer function H(f)=S_{ηa}(f)/S_{aa}(f) and coherence γ²(f) over f∈[0.5,50] Hz. Direct forcing leakage is excluded only if |H(f)| and γ²(f) are indistinguishable from the statistical noise floor at every f < f_p while the f^-1 tail persists. Complementary check: drive the shaker with a pure 75 Hz sinusoid; if the low-frequency tail still appears, it cannot be a broadband forcing artifact, whereas if it disappears, the tail is directly forced. Report both spectra with uncertainty bands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The claim that large scales reach Rayleigh-Jeans equilibrium rests on the assumption that energy in the band f_L < f < f_p arrives solely through nonlinear three-wave interactions from the 50-100 Hz forcing. If the shaker, wavemaker, or water column also radiates a low-frequency component (mechanical subharmonics, membrane coupling, boundary effects), then the observed k^-1 and f^-1 spectra, Gaussian statistics, and near-zero flux could be those of a directly driven random field rather than a thermalized one. The authors give circumstantial evidence: a solid ring suppresses square-tank eigenmodes (Experimental setup), large-scale dissipation is reported below 5%, and zero net flux is shown in the Fig. 4 inset. But they never report the accelerometer spectrum below f_p, nor the coherence between the drive signal and the large-scale wave field, and the flux estimate is an indirect dissipation-based quantity with no error bars. Because f_L lies at the lower end of the fitted decade, even a weak low-frequency mechanical tail could anchor the spectrum. This is the weakest link between the measurement and the equilibrium interpretation; the thermodynamic derivatives in the Entropy and heat capacity section inherit it because they are computed from the same band.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports an experiment on hydroelastic wave turbulence in a square tank covered by a thin elastic sheet, forced randomly at small scales (50–100 Hz) by a circular wavemaker. The authors measure the wave amplitude in space (Fourier transform profilometry) and time (laser Doppler velocimetry) and observe that, at scales larger than the forcing scale, the spatial spectrum scales as k^-1 and the frequency spectrum scales as f^-1/T, in agreement with the predicted Rayleigh–Jeans equilibrium spectra of hydroelastic tensional waves. They also report a Gaussian amplitude distribution, zero net energy flux in the equilibrium band, an effective temperature of order 10^15 K, and entropy and heat-capacity estimates that follow the equipartition prediction C_v = N_f k_B/2. The central claim is the first experimental evidence of statistical equilibrium of the large scales of hydroelastic wave turbulence, with the wave field behaving like a thermalized collection of Fourier modes over more than a decade.","tokens_in":9029,"tokens_out":3872,"duration_ms":49597,"significance":"If the interpretation is correct, the paper would provide a clear experimental realization of large-scale statistical equilibrium in a wave-turbulence system that is resolved in both space and time, extending earlier single-point observations in capillary and flexural wave turbulence. The dispersion-relation collapse across tension values in Fig. 2 and the f^-1/T rescaling in Fig. 4 are visually compelling, and the zero-flux measurement is a relevant control. The quantitative thermodynamics (effective temperature, entropy, heat capacity) is a natural extension, though its evidential weight is weaker because it relies on the same spectral assumption used to extract the temperature. The main risk is that the equilibrium attribution depends on the absence of direct low-frequency forcing from the wavemaker, a point that is asserted but not directly measured in the manuscript.","major_comments":[{"comment":"The attribution of the band f_L < f < f_p to nonlinear thermalization rests on the assumption that the shaker/wavemaker does not directly inject energy into those scales. The manuscript reports neither the accelerometer spectrum below f_p nor the coherence between the drive signal and the large-scale wave field, and it does not provide an off-resonance control (for example, shifting the forcing band while leaving the low-frequency mechanical response unchanged). Since f_L lies at the lower edge of the fitted decade, even a weak low-frequency mechanical tail or a residual tank eigenmode could produce the reported k^-1 and f^-1 spectra without implying thermalization. Please provide a quantitative transfer-function-style test or an explicit bound on direct low-frequency forcing.","section":"Experimental setup; Equilibrium power spectra (Fig. 4 inset)"},{"comment":"The central quantitative claim is the k^-1 and f^-1/T scaling over more than a decade, but no fitting procedure, fit ranges, slope values, or confidence intervals are reported. Please give the fitted slopes with uncertainties and demonstrate insensitivity to the endpoints f_L and f_p; otherwise the 'over more than a decade' statement is difficult to evaluate.","section":"Equilibrium power spectra (Figs. 3 and 4)"},{"comment":"The effective temperature theta is obtained by integrating the measured spectrum under the assumed f^-1 form of Eq. (4), and the subsequent verification that theta scales as sigma_eta^2 follows from Parseval's theorem applied to the same spectrum. This is therefore an internal consistency check rather than an independent prediction. Likewise, the entropy derivative and heat capacity are computed by injecting the Rayleigh-Jeans spectrum, so the agreement in Fig. 6 is a consistency test of the equilibrium ansatz rather than a new thermodynamic statement. Please reframe these sections accordingly and avoid presenting them as independent verifications.","section":"Effective temperature; Entropy and heat capacity (Eq. (5), Fig. 5)"},{"comment":"The zero net energy flux is a load-bearing observable, but the inset reports it without error bars and with only a brief reference to a dissipation-based method in the Supplemental Material. Given that this measurement distinguishes equilibrium from directly driven random fields, please include uncertainties and a concise description of the dissipation model used to infer the flux.","section":"Equilibrium power spectra (Fig. 4 inset)"}],"minor_comments":[{"comment":"The symbol L is used first for the tank side (600 mm) and later as 'the ring diameter' in the definition of S_eta(k,omega); please define the ring diameter explicitly and distinguish it from the tank dimension.","section":"Dispersion relation (spectrum definition)"},{"comment":"The phrase 'first axisymmetrical eigenmode' should be 'axisymmetric', and the value f_L = 1.4 Hz should be identified as the eigenmode of the tank, of the ring, or of the sheet-covered system, since this is the lower bound of the fitted decade.","section":"Equilibrium power spectra; Fig. 4 caption"},{"comment":"In the sentence on bending waves, the notation '1/ℓ_tb ~ 300 m^-1' is inconsistent with the definition 'ℓ_tb = 2π sqrt(B/T)'; please clarify whether the quoted value is 1/ℓ_tb or a typographical error.","section":"Theoretical predictions (Eq. (2))"},{"comment":"The sentence 'Note that the sheet at rest is flat everywhere except close to the solid ring where all the curvature is confined' suggests the sheet is not perfectly flat in the measurement region near the ring; please state whether the reported data exclude that curved region.","section":"Experimental setup"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the data appear carefully collected, but the equilibrium claim is currently supported mainly by spectral shape and consistency checks. The most important missing piece is a direct test of whether the forcing leaks into the large-scale band; without that, the 'statistical equilibrium' attribution is not fully established. I would be more comfortable with a version that adds an accelerometer-based low-frequency spectrum or drive-signal coherence measurement, fit uncertainties on the slopes, and error bars on the flux."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper reports the first experiment claiming statistical equilibrium of large-scale hydroelastic waves with both spatial and frequency spectra, and that claim mostly holds. The k^-1 and f^-1/T scalings over a decade, the collapse across tensions, Gaussian amplitude statistics, and zero net flux measured in the equilibrium band add up to a convincing picture. It extends prior capillary/plate equilibrium work to a new system with spatiotemporal resolution, and it is honestly positioned as an extension rather than a revolution.\n\nWhat I like: the dispersion relation collapse with no fitted parameter (though T is inferred), the use of the solid ring to kill tank eigenmodes, and the flux measurement with a reference experiment for large-scale forcing. The paper also states its own limitations implicitly—the thermodynamic temperature and heat capacity estimates are internal consistency checks. The heat capacity result C_v ≈ N_f k_B/2 follows from the spectrum you already measured, so it is not an independent test of thermodynamics. The authors don't oversell this, but the abstract says 'demonstrating that classical thermodynamic concepts apply,' which is a step beyond what the data show. That's a wording problem more than a scientific one.\n\nThe real soft spot is the one the stress-test flags: the equilibrium interpretation depends on large-scale energy arriving via nonlinear interactions, not directly from the wavemaker. The paper shows the spectra, the zero flux, and low large-scale dissipation, but it never reports the accelerometer spectrum below the forcing band or the coherence between drive and large-scale wave field. A weak low-frequency mechanical tail could anchor the spectrum. I don't think this kills the paper—random forcing at 50–100 Hz with a small wavemaker is unlikely to inject coherent large-scale energy, and the Gaussian statistics plus zero flux are consistent with thermalization—but the authors should provide the transfer-function check or at least the low-frequency accelerometer spectrum. It would close the only real loophole.\n\nI also want error bars on the spectral slopes. The fits look good, but one decade with no uncertainties is thin for a claim of exact -1.\n\nBottom line: this deserves peer review. The central result is likely right, the experiment seems careful, and the loophole is fixable with data they probably already have. I'd send it out.","headline":"Solid first spatiotemporal evidence for large-scale Rayleigh-Jeans equilibrium in hydroelastic waves; main gap is ruling out direct low-frequency forcing leakage.","tokens_in":9572,"tokens_out":1425,"would_cite":true,"duration_ms":16315,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper reports the first experimental evidence that large-scale hydroelastic wave turbulence can settle into a statistical-equilibrium, Rayleigh-Jeans thermalized state while the small scales remain driven and dissipative.","keywords":["hydroelastic waves","wave turbulence","statistical equilibrium","Rayleigh-Jeans spectrum","equipartition","effective temperature","energy flux","tensional waves"],"falsifier":"Measure the large-scale spectrum while reducing the forcing amplitude until wave-wave interactions are negligible: if the $k^{-1}$ and $f^{-1}$ shapes persist with amplitudes set by the forcing level rather than by the effective temperature, the apparent equilibrium is a forced tail. Alternatively, a direct measurement of the nonlinear energy transfer in the band below the forcing scale, for example from third-order correlations, showing a nonzero net flux would contradict the zero-flux thermalization claim.","tokens_in":8589,"feed_emoji":"🌊","tokens_out":8814,"duration_ms":97687,"temperature":0.7,"pith_summary":"This paper reports the first experimental evidence that the large scales of hydroelastic wave turbulence can sit in a thermalized, statistical-equilibrium state while the small scales remain driven and dissipative. The experiment forces random waves at small scales on a water surface covered by a stretched elastic sheet and observes the much larger scales in space and time. In that large-scale band the measured spectra follow the Rayleigh-Jeans equilibrium predictions, $S_\\eta(k)\\propto k^{-1}$ and $S_\\eta(f)\\propto f^{-1}$, over more than a decade, with zero net energy flux, Gaussian wave-amplitude statistics, and Boltzmann-like energy fluctuations. If the claim holds, the large scales of a driven turbulent wave system can be described by classical thermodynamics, with an effective temperature, an entropy, and a heat capacity, rather than by a cascade of energy through scales.","feed_headline":"Turbulent waves thermalize above the forcing scale","feed_subtitle":"Large-scale modes of a driven hydroelastic wave field reach equilibrium with zero net energy flux.","key_machinery":"The central object is the Rayleigh-Jeans equilibrium spectrum of weak wave turbulence, the wave analogue of the blackbody spectrum. Because hydroelastic waves interact through three-wave processes, wave action is not conserved and no inverse cascade carries flux toward large scales, so modes larger than the forcing scale are expected to equipartition, giving $E^{Eq}(k)=k_B\\theta/(2\\pi\\rho)k$ and hence the measured $S_\\eta(k)\\propto k^{-1}$ and $S_\\eta(f)\\propto f^{-1}$. The experimental machinery that exposes this regime is a square tank with a silicone sheet, a solid ring that suppresses tank eigenmodes, random small-scale forcing from a circular wavemaker, space-time resolved Fourier transform profilometry, and an energy-flux estimate built from spectral dissipation. The dispersion relation $\\omega^2=gk+(T/\\rho)k^3+(B/\\rho)k^5$ connects the spatial and frequency spectra, and the tension-dominated nature of the waves is verified by the collapse of $f/\\sqrt{T}$ against $k^{3/2}$.","core_discovery":"On its own terms, the paper establishes that hydroelastic waves on a silicone sheet forced randomly at 50 to 100 Hz thermalize at scales larger than the forcing scale: the wave field's spatial and temporal power spectra agree with the Rayleigh-Jeans equilibrium spectra $S_\\eta^{Eq}(k)=k_B\\theta/(2\\pi T)k^{-1}$ and $S_\\eta^{Eq}(f)=k_B\\theta/(3\\pi T)f^{-1}$ over more than a decade of scales, down to the first tank eigenmode. The measured energy flux in this band is zero, large-scale dissipation is below 5%, and the integrated spectrum gives an effective temperature $\\theta\\simeq 8\\times10^{15}$ K, with the entropy derivative and heat capacity matching equipartition among $N_f\\simeq 5600$ Fourier modes. The authors frame this as the coexistence of equilibrium large scales with nonequilibrium small-scale dynamics, made visible by suppressing tank eigenmodes with a solid ring and by resolving the wave field in both space and time.","pith_inferences":["If the three-wave thermalization mechanism is generic, then any wave-turbulence system without an inverse cascade should exhibit a measurable Rayleigh-Jeans band above the forcing scale; applying the same space-time method to gravity-capillary or pure flexural waves would test that universality.","The inferred temperature rests on the integrated spectral amplitude, so an independent check would be to force with two well-separated small-scale bands and test whether the two effective temperatures add like equilibrium heat baths.","A per-mode heat capacity equal to $k_B$ suggests the equilibrium band is an ideal classical wave gas; lowering the forcing cutoff could drive it toward condensation-like effects, analogous to classical wave condensation in optics."],"forward_implications":["Above the forcing scale, wave-turbulence spectra should be expected to have an equilibrium, Rayleigh-Jeans form rather than a cascade form, at least in systems without an inverse cascade.","Forcing strength acts as a temperature knob: the effective temperature grows with the wave-amplitude variance, $\\theta\\sim\\sigma_\\eta^2$, so the equilibrium band can be tuned experimentally.","Thermodynamic response functions, including entropy and a per-mode heat capacity equal to $k_B$, become measurable quantities for a turbulent wave field.","The coexistence of equilibrium large scales and nonequilibrium small scales sets up a system in which tools of equilibrium and nonequilibrium statistical mechanics, such as fluctuation relations, can be tested experimentally.","For ice-covered oceans, where swells can act as small-scale forcing, large-scale flexural-gravity waves may be described by an effective temperature rather than by cascade scaling."],"supporting_citations":[{"why":"Supplies the weak-turbulence theory of statistical equilibrium, Rayleigh-Jeans spectra, and the H-theorem used to identify the equilibrium band.","marker":"[7]"},{"why":"Provides the prediction that large scales of three-wave wave-turbulence systems reach equipartition, giving the equilibrium spectral density.","marker":"[9]"},{"why":"Prior experimental observation of thermal equilibrium in capillary wave turbulence, the single-point precedent this work extends to space-time resolved hydroelastic waves.","marker":"[10]"},{"why":"Recent experimental evidence of statistical equilibrium of large scales in 3D hydrodynamic turbulence, used as a comparison for the effective-temperature scale.","marker":"[5]"},{"why":"Establishes the hydroelastic wave-turbulence experiment and dispersion relation for a silicone-sheet-covered water surface on which this setup builds.","marker":"[22]"},{"why":"Provides experimental evidence of hydroelastic three-wave interactions, the nonlinearity that suppresses inverse cascade and enables thermalization.","marker":"[23]"},{"why":"Supplies the method for measuring the spectral energy flux from dissipated energy, used to report zero net flux in the equilibrium band.","marker":"[39]"},{"why":"Provides the Fourier transform profilometry technique used for the spatially and temporally resolved wave-field measurement.","marker":"[38]"}],"fun_headline_variants":["Turbulent waves settle into thermal equilibrium at large scales","Large-scale waves thermalize despite random forcing","Zero energy flux at large scales of wave turbulence","Statistical equilibrium emerges in driven wave turbulence","Turbulence large scales behave like a thermal bath"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the small circular wavemaker forcing at 50 to 100 Hz does not directly inject energy into scales larger than the forcing scale, so the observed $k^{-1}$ and $f^{-1}$ spectra and the zero net flux reflect genuine nonlinear thermalization rather than a low-frequency tail of the forcing or residual tank eigenmodes.","fun_headline_variants_meta":{"raw":{"variants":["Turbulent waves settle into thermal equilibrium at large scales","Large-scale waves thermalize despite random forcing","Zero energy flux at large scales of wave turbulence","Statistical equilibrium emerges in driven wave turbulence","Turbulence large scales behave like a thermal bath"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001087,"raw_usage":{"total_tokens":4505,"prompt_tokens":866,"completion_tokens":3639,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":482,"completion_tokens_details":{"reasoning_tokens":3568}},"tokens_in":482,"tokens_out":3639,"duration_ms":30977,"temperature":1.0,"reasoning_tokens":3568,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:05:39.294886+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the large-scale spectrum while reducing the forcing amplitude until wave-wave interactions are negligible: if the $k^{-1}$ and $f^{-1}$ shapes persist with amplitudes set by the forcing level rather than by the effective temperature, the apparent equilibrium is a forced tail. Alternatively, a direct measurement of the nonlinear energy transfer in the band below the forcing scale, for example from third-order correlations, showing a nonzero net flux would contradict the zero-flux thermalization claim.","supporting_citations":[{"cited_title":"Balkovsky, G","cited_arxiv_id":null,"evidence_quote":"Provides the prediction that large scales of three-wave wave-turbulence systems reach equipartition, giving the equilibrium spectral density."},{"cited_title":"Michel, F","cited_arxiv_id":null,"evidence_quote":"Prior experimental observation of thermal equilibrium in capillary wave turbulence, the single-point precedent this work extends to space-time resolved hydroelastic waves."},{"cited_title":"Gorce and E","cited_arxiv_id":null,"evidence_quote":"Recent experimental evidence of statistical equilibrium of large scales in 3D hydrodynamic turbulence, used as a comparison for the effective-temperature scale."},{"cited_title":"Deike, J","cited_arxiv_id":null,"evidence_quote":"Establishes the hydroelastic wave-turbulence experiment and dispersion relation for a silicone-sheet-covered water surface on which this setup builds."},{"cited_title":"Deike, M","cited_arxiv_id":null,"evidence_quote":"Provides experimental evidence of hydroelastic three-wave interactions, the nonlinearity that suppresses inverse cascade and enables thermalization."},{"cited_title":"Deike, M","cited_arxiv_id":null,"evidence_quote":"Supplies the method for measuring the spectral energy flux from dissipated energy, used to report zero net flux in the equilibrium band."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Fourier transform profilometry technique used for the spatially and temporally resolved wave-field measurement."}],"review_version":1}