{"id":"77d8e420-87f2-4a66-bb79-991b347aedd5","arxiv_id":"2607.01404","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Closed-form first-passage-time distribution for an underdamped harmonic oscillator (short-time Hamiltonian + long-time Kramers) agrees with micro-cantilever data and yields the power of an information engine.","lead":"The paper derives the full first-passage-time distribution for an underdamped harmonic oscillator by combining a short-time Hamiltonian approximation with Kramers eigenvalues for longer times, and matches it to micro-cantilever experiments. The result is used to compute and verify the power of an information engine that extracts work from first-passage events.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged Q-dependence of the short-time Hamiltonian construction.","rationale":"The strongest claim is that Eq. 2 supplies a closed-form, experimentally verified FPT pdf for the underdamped oscillator. The only non-trivial modeling step that could undermine that claim is the use of the dissipationless phase-space measure for t_fp≤2π at moderate Q. The reader already identified this step. Direct inspection of the manuscript shows that the same experimental data used to validate the long-time Kramers rate also validate the short-time plateau and the instantaneous weight to high precision (Figs. 1b, 3). Because that empirical agreement is already present, the theoretical soft spot does not propagate into a correctness risk for the published claim. No additional internal inconsistency, missing derivation, or experimental mismatch appears. The verdict therefore remains CONDITIONAL (pending public data and the companion spectral details) with no further adjustment required.","tokens_in":9897,"tokens_out":573,"duration_ms":5379,"concrete_test":"Extract the plateau height of P(t_fp) for 0<t<π from the already-acquired micro-cantilever time series at the three reported B values and compare it to (e^{-B})/(2π); if the relative discrepancy remains <5% (as Fig. 3 already suggests), the Hamiltonian short-time construction is empirically validated at Q=7 and the closed-form claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly isolates the softest point: after Eq. 12 the authors assert that the short-time pieces P_I and P_II (derived from the purely Hamiltonian measure P(\theta_{0},E)=e^{-E}/(2π) and the cutoff E†(t)) \"shouldn't depend strongly on Q,\" even though the experimental Q=7 makes the energy-relaxation time only a few periods and the phase-space frontiers are acknowledged to be porous. That premise is load-bearing for the closed-form claim of Eq. 2, yet the paper itself already supplies the decisive empirical check: Fig. 1(b) and Fig. 3 show that both the plateau height of P_II and the instantaneous weight P_I match the Hamiltonian formulas to within statistical error at precisely this Q. The companion spectral calculation and the ancillary Langevin movies further corroborate that residual dissipation inside one period does not shift the short-time pdf at a level that would invalidate the formula. Consequently the concern, while real in principle, does not land as a material threat to the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript derives a closed-form expression for the first-passage-time pdf of an underdamped harmonic oscillator to a position threshold x_B. The distribution is decomposed as P(t_fp)=P_I δ(t_fp)+P_II(t_fp)+P_III(t_fp) (Eq. 2), where P_I is the equilibrium probability of already lying beyond the barrier, P_II is obtained from a Hamiltonian phase-space construction with a time-dependent energy cutoff E†(t) (Eqs. 7–12), and P_III is constructed from a time-dependent Kramers escape rate Γ(t)=λ_1[E†(t)] that becomes constant for t>2π. The formula is tested against high-resolution interferometric measurements on a micro-cantilever (Q≃7) and is used to predict the mean power of an information engine that extracts work at first passage; both the pdf and the engine power agree quantitatively with experiment.","tokens_in":10121,"tokens_out":951,"duration_ms":8244,"significance":"A usable closed-form FPT distribution for underdamped oscillators has been missing; the only previously known closed form is the long-time free-particle (random-acceleration) result. The present construction supplies an explicit, parameter-light expression that covers the full time axis and is validated by direct experiment at moderate Q. The information-engine application further demonstrates that the distribution is immediately useful for quantitative predictions in stochastic thermodynamics. The combination of analytic formulae, spectral rates, ancillary Langevin movies, and independent experimental checks constitutes a solid advance for the field.","major_comments":[{"comment":"After Eq. 12 the authors assert that the short-time pieces P_I and P_II, derived from the purely Hamiltonian measure P(θ_0,E)=e^{-E}/(2π), “shouldn’t depend strongly on Q” even though the experimental Q=7 makes the energy-relaxation time only a few periods. While Figs. 1(b) and 3 show that the plateau height and instantaneous weight match the Hamiltonian formulae at this particular Q, the manuscript itself does not quantify residual dissipation inside one period. A short statement (or a reference to the companion spectral calculation) that the relative error remains below the experimental uncertainty for Q≳ few would make the closed-form claim fully self-contained.","section":null}],"minor_comments":[{"comment":"The companion paper [22] is cited for the full eigenvalue derivation of λ_1 and for the large-Q limit; a one-sentence sketch of how λ_1 is obtained (or an explicit formula for the constant rate Γ_B) would help readers who do not immediately consult the companion.","section":null},{"comment":"Fig. 2 phase-space cartoons are dense; labeling the four panels more prominently (e.g., “t=0”, “0<t<π”, …) and adding a brief caption sentence that the movies are available as ancillary files would improve readability.","section":null},{"comment":"In Eq. (16) the average work contains a term proportional to L(1/√(2π)e^{-B}-1); a short remark clarifying that this term arises solely from the instantaneous-trigger contribution would avoid possible confusion with the plateau of P_II.","section":null},{"comment":"Typographical: “APPLICA TION” and “INFORMA TION” in the section heading contain spurious spaces; “an” should be “and” in the abstract sentence on engine power.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The central claim is sound and the experimental agreement is convincing. The only load-bearing soft spot (Q-independence of the short-time Hamiltonian construction) is already checked empirically at the working Q; a brief quantitative remark would convert the paper into a clean accept. Fit for a Letter-style journal is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new result here is a usable three-piece closed-form pdf for first-passage time of an underdamped harmonic oscillator: instantaneous weight from the equilibrium position measure, a short-time plateau from Hamiltonian phase-space flow, and a long-time exponential from the slowest Kramers eigenvalue (or energy-diffusion rate). That object was missing; free-particle long-time asymptotics and overdamped formulas do not cover it. They then use the same distribution to predict information-engine power and check both against a Q≃7 micro-cantilever.\n\nWhat works: the construction is transparent, the formulas are explicit (P_I = ½ erfc√B, P_II = (e^{-B}-e^{-E†(t)})/2π, Γ(t)=λ₁[E†(t)]), and the experiment is clean. Figs. 1b and 3 show quantitative agreement for the delta weight, the plateau height, and the long-time rate across several barriers; Fig. 4 matches engine power versus L and x_B. No free parameters are fitted to the histograms; B and Q are measured. The companion paper and Langevin movies are the right place for the spectral details and the Q-dependence scans.\n\nSoft spot, already flagged and already tested: after Eq. 12 they claim the short-time pieces “shouldn’t depend strongly on Q” even though at Q=7 the energy-relaxation time is only a few periods and the phase-space frontiers are porous. That is load-bearing for the closed form. Empirically it holds—the plateau and P_I sit on the Hamiltonian curves within error—so the concern does not sink the claim, but a reader will want the companion’s Q-scans before treating the formula as universal. Data are promised but not yet public; that is minor.\n\nThis is for people who need FPT statistics or power estimates in underdamped stochastic thermodynamics and micro-mechanics. The math is standard Kramers plus phase-space bookkeeping, the citations are appropriate, and the experimental protocol is reproducible. I would send it to referees; the central result is solid enough to deserve that time. Worth reading if you work in the area; I would cite the distribution when I next need an underdamped FPT formula.","headline":"Closed-form underdamped FPT pdf that actually matches cantilever data and engine power; the Q=7 Hamiltonian short-time piece is the soft spot but is already checked by the figures.","tokens_in":10764,"tokens_out":579,"would_cite":true,"duration_ms":5868,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"The first-passage time distribution of an underdamped harmonic oscillator is obtained in closed form by combining short-time Hamiltonian phase-space orbits with Kramers eigenvalues, matching micro-cantilever data and predicting information-","keywords":["first passage time","underdamped harmonic oscillator","Kramers operator","information engine","energy diffusion","micro-cantilever","stochastic thermodynamics"],"falsifier":"Record the short-time plateau height of the first-passage density on resonators whose quality factors range from a few units to several hundred; systematic depression of the plateau below e^{-B}/(2π) as Q falls would falsify the Hamiltonian short-time claim.","tokens_in":10786,"feed_emoji":"⏳","tokens_out":889,"duration_ms":20007,"temperature":0.7,"pith_summary":"This paper derives the full probability distribution of the first time an underdamped harmonic oscillator reaches a fixed position threshold. Short times are handled by a Hamiltonian approximation that tracks which initial energies and phases hit the barrier within one oscillation; longer times use the slowest eigenvalue of the Kramers operator (or energy-diffusion rates at high quality factor). The resulting three-term formula depends only on barrier height and quality factor, agrees quantitatively with high-resolution experiments on a micro-cantilever, and supplies the mean waiting time needed to compute the power of an information engine that extracts work on each first crossing. The result fills a long-standing gap for systems with inertia, where the usual overdamped methods fail.","feed_headline":"Closed-form first-passage times for underdamped oscillators","feed_subtitle":"Theory matches micro-cantilever data and predicts the power of engines that harvest barrier crossings","key_machinery":"The three-term decomposition P(t_fp)=P_I δ(t_fp)+P_II(t_fp)+P_III(t_fp), in which P_II is obtained by integrating the equilibrium density e^{-E}/(2π) over the Hamiltonian phase-space region that reaches the barrier in time t_fp, and P_III is built from the time-dependent escape rate equal to the slowest Kramers eigenvalue at the instantaneous energy frontier E†(t).","core_discovery":"The first-passage-time density for position of an underdamped harmonic oscillator is exactly the sum of three pieces: a Dirac delta of weight one-half erfc of the square root of the barrier for instantaneous crossings, a short-time plateau-plus-cutoff obtained by integrating the equilibrium measure over Hamiltonian orbits that hit the barrier inside one period, and a long-time exponential tail whose rate is the leading eigenvalue of the Kramers operator evaluated at a time-dependent energy threshold. The expression is fully determined by barrier height B and quality factor Q and is confirmed by experiment.","pith_inferences":["The same phase-space construction can be extended to weakly anharmonic wells by replacing circular orbits with the closed energy contours of the actual potential.","Deviation of the measured short-time plateau from the Hamiltonian prediction at intermediate Q would furnish a direct experimental measure of energy diffusion per cycle.","The analytic density supplies a clean benchmark for numerical solvers of the two-dimensional Fokker-Planck equation with absorbing boundaries."],"forward_implications":["Mean first-passage times and higher moments for underdamped resonators become available without stochastic simulation.","Power of information engines that harvest work at first crossings can be optimized analytically over threshold and stroke length.","The short-time plateau height directly reports the equilibrium probability of super-threshold initial energies.","Long-time rates recover classic Kramers escape while the full density interpolates toward free-particle random-acceleration statistics as the restoring force vanishes."],"fun_headline_variants":["Exact FPT density for underdamped oscillators: delta plus short-time plus Kramers tail","Underdamped first-passage times solved in closed form and verified on micro-cantilever","Three-piece FPT for underdamped HO matches experiment and predicts info-engine power","First-passage density of underdamped oscillator fixed by barrier height and quality factor","Closed-form first-passage times for underdamped oscillators confirmed experimentally"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The short-time plateau and cutoff can be computed from purely Hamiltonian orbits even at moderate quality factor, as if energy does not diffuse appreciably inside a single oscillation period.","fun_headline_variants_meta":{"raw":{"variants":["Exact FPT density for underdamped oscillators: delta plus short-time plus Kramers tail","Underdamped first-passage times solved in closed form and verified on micro-cantilever","Three-piece FPT for underdamped HO matches experiment and predicts info-engine power","First-passage density of underdamped oscillator fixed by barrier height and quality factor","Closed-form first-passage times for underdamped oscillators confirmed experimentally"]},"model":"grok-4.5","effort":"low","cost_usd":0.006682,"raw_usage":{"total_tokens":1645,"prompt_tokens":701,"num_sources_used":0,"completion_tokens":114,"cost_in_usd_ticks":66820000,"prompt_tokens_details":{"text_tokens":701,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":830,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":701,"tokens_out":114,"duration_ms":7098,"temperature":1.0,"reasoning_tokens":830,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T08:51:08.895859+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Record the short-time plateau height of the first-passage density on resonators whose quality factors range from a few units to several hundred; systematic depression of the plateau below e^{-B}/(2π) as Q falls would falsify the Hamiltonian short-time claim.","supporting_citations":[],"review_version":2}