{"id":"4f601816-dc1c-4ad4-ad26-4ceab6df1e39","arxiv_id":"2607.01405","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The first-passage-time distribution of an underdamped harmonic oscillator is obtained analytically for short, intermediate and long times across quality factors and matches Langevin simulations.","lead":"The paper derives the full first-passage-time distribution for an underdamped harmonic oscillator to cross a position threshold, using quality-factor-dependent methods. The result closes a gap in inertial first-passage theory and matches simulations, with direct use for underdamped information engines.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the short-time rate ansatz as the sole uncontrolled approximation and correctly notes that it is confined to intermediate times and does not undermine the long-time eigenvalue or energy-diffusion results. Cross-checks already present in the manuscript (Figs. 2–3, 6–7, ancillary movies, and the companion experimental letter) give independent confirmation of every other piece of the decomposition. Because that residual uncertainty is both localized and openly acknowledged, no adjustment of the ACCEPT verdict is warranted.","tokens_in":13371,"tokens_out":418,"duration_ms":4869,"concrete_test":"Extract the first-passage histograms already generated for Q=7 and Q=15 (Appendix A) and recompute the intermediate-time density using the pure long-time rate λ₁(B) instead of λ₁(E†(t)). If the L1 discrepancy on [π,2π] remains below a few percent of the peak height, the ansatz is numerically harmless; a larger discrepancy would quantify the only residual uncertainty without altering the long-time or large-Q results.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that the FPT density decomposes into the three explicit pieces (3a–c) with Γ given by λ₁ of the absorbing Kramers operator (moderate Q) or by 1/(π+τ_Z) (large Q)—is supported by independent numerical diagonalization of L_xB, direct Langevin sampling of the same rate, and the closed-form energy-diffusion mean time (18)–(20). The only uncontrolled step is the short-time ansatz Γ(t)=λ₁(E†(t)) for π<t<2π (Sec. VI). That ansatz is clearly labeled, is used only for intermediate times, and does not affect either the long-time exponential or the large-Q limit. No internal inconsistency or hidden assumption that would invalidate the claimed formulas was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript derives the first-passage-time distribution P(t_fp) for the position of an underdamped harmonic oscillator to reach a threshold x_B. The density is decomposed into three contributions (Eqs. 2–3): an instantaneous Dirac piece P_I from initial conditions already above threshold, a short-time Hamiltonian flux P_II of high-energy initial conditions (t_fp ≤ 2π), and a long-time Kramers-escape piece P_III. For moderate Q the long-time rate is identified with the slowest eigenvalue λ_1 of the absorbing Kramers operator L_xB (Sec. III); for large Q it is obtained from the mean first-passage time of the energy-diffusion process plus a half-period delay (Sec. IV, Eqs. 18–26). Short-time dynamics are treated in the Hamiltonian approximation. Direct Langevin simulations, numerical diagonalization of L_xB, and the closed-form energy-diffusion time all agree. The mean trajectories conditioned on first passage are shown to follow an instanton/relaxation path driven by a characteristic noise pattern (Sec. V).","tokens_in":13519,"tokens_out":894,"duration_ms":8644,"significance":"A quantitative FPT distribution for underdamped oscillators has been missing; most prior analytic work addresses the overdamped limit. The paper supplies closed-form expressions that cover the full range of Q and t_fp, validated by independent numerical methods (eigenvalue spectra of L_xB, long Langevin trajectories, energy-diffusion MFPT) with no free parameters. The companion Letter already uses the mean FPT for an information-engine power estimate that matches experiment; the present article supplies the technical foundation and extends the result to very large Q. The identification of the noise pattern that generates escape events is an additional, potentially reusable insight. These results are of direct interest for stochastic thermodynamics, underdamped information engines, and any setting in which inertial first-passage statistics matter.","major_comments":[],"minor_comments":[{"comment":"Sec. VI: the short-time rate ansatz Γ(t)=λ_1(E†(t)) for π<t<2π is clearly labeled an educated guess and works well for Q=7, but a one-sentence quantitative statement of residual discrepancy (or a brief comparison against the full spectrum of L_xB) would help readers judge its accuracy for intermediate Q.","section":null},{"comment":"Fig. 7 and the accompanying text note the plateau structure of the transient for large Q; a short remark that these plateaux arise from the imaginary parts of the higher eigenvalues of L_xB (already mentioned in Sec. VI) would make the connection fully explicit.","section":null},{"comment":"Eq. (5) and the surrounding paragraph: the rough estimate Γ_K=e^{-B}/Q is useful, but a brief cross-reference to the more accurate expressions Γ_B=λ_1 and Γ_Z later in the paper would avoid any impression that Γ_K is the final result.","section":null},{"comment":"Appendix A: the reconstruction of the FPT histogram from residence times is elegant; a single sentence clarifying that the same long trajectory can be re-analyzed for any x_B would further emphasize the numerical efficiency.","section":null},{"comment":"Minor typographical points: “tfp” vs “t_fp” consistency, and a few missing spaces after commas in the abstract and Sec. II.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid technical companion to the already-submitted Letter. The central claims are independently cross-checked by three numerical routes and contain no free parameters. The only uncontrolled step (the short-time rate ansatz) is local, clearly labeled, and does not affect the long-time or large-Q results. I see no reason to delay publication; minor polishing can be handled at the proof stage if desired."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finally gives a usable, Q-dependent first-passage-time density for the underdamped harmonic oscillator. That was a real gap; most of the literature stops at overdamped or mean times. They split P(t_fp) into three pieces: instantaneous mass above threshold, short-time Hamiltonian flux of high-energy initial conditions, and long-time survival driven by a Kramers rate. For moderate Q the rate is the slowest eigenvalue of the absorbing Kramers operator; for large Q it is the energy-diffusion mean first-passage time plus half a period. Both are derived cleanly and checked against independent Langevin runs and FEM diagonalization (Figs. 2–3). Agreement is excellent, no free parameters.\n\nThe short-time Hamiltonian part and the large-Q energy-diffusion formula look airtight. The mean escape trajectories and the associated noise pattern (instanton + free relaxation) are a nice extra that matches the data without tuning. The companion letter already uses the mean FPT for an information engine, so the result is not purely formal.\n\nThe only soft spot is the intermediate-time ansatz that replaces the time-dependent rate by λ₁ evaluated at the instantaneous contact energy E†(t). They call it an educated guess and it is used only for π < t < 2π. It works well at Q ~ 7; at larger Q the plateaus from higher eigenvalues remain visible and are not analytically captured. That is a genuine limitation, but it is clearly labeled and does not touch the long-time exponential or the large-Q limit.\n\nMath is standard stochastic calculus, citations are appropriate, and they promise public data and movies. This is for people who actually need the distribution (resonators, underdamped engines, sensors), not for a broad conceptual rewrite. I would send it to referees without hesitation; the core claims hold.","headline":"Solid, usable FPT density for underdamped oscillators: three explicit pieces, spectral and energy-diffusion rates, clean numerics; one labeled short-time ansatz.","tokens_in":14123,"tokens_out":482,"would_cite":true,"duration_ms":4758,"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 of an underdamped harmonic oscillator splits into three pieces fixed by Q, energy diffusion, and Kramers eigenvalues.","keywords":["first passage time","underdamped oscillator","quality factor","Kramers equation","energy diffusion","absorbing boundary","instanton","noise pattern"],"falsifier":"A long Langevin trajectory (or experiment) at moderate Q whose first-passage histogram fails to match the three-piece formula, especially the height and duration of the early plateaus controlled by the instantaneous contact energy E†(t).","tokens_in":14285,"feed_emoji":"⏱️","tokens_out":708,"duration_ms":5812,"temperature":0.7,"pith_summary":"This paper gives a closed-form distribution for the first time an underdamped harmonic oscillator, started from equilibrium, reaches a fixed position threshold. Because the oscillator has inertia, the answer depends strongly on the quality factor Q and cannot be taken from the well-studied overdamped case. The authors split the density into an instantaneous piece (initial conditions already above threshold), a short-time piece controlled by pure Hamiltonian rotation of high-energy states, and a long-time piece whose exponential rate is the slowest eigenvalue of the Kramers operator with an absorbing wall (or, for very large Q, the mean energy-crossing time plus half a period). Direct Langevin simulations confirm the formulas across a wide range of Q and barrier heights. They also show that every trajectory that hits the threshold is driven by a characteristic, resonantly amplified noise pattern whose average shape is known exactly from time-reversal of the free relaxation.","feed_headline":"Underdamped first-passage times split into three Q-dependent pieces","feed_subtitle":"Hamiltonian flux, Kramers eigenvalues and energy diffusion give the full density, matching simulations","key_machinery":"The decomposition P(t_fp)=P_I δ(t_fp)+P_II(t_fp)+P_III(t_fp), with P_III built from the slowest eigenvalue λ_1 of the Kramers operator L_xB that includes an absorbing sink at x=x_B (or, equivalently, from the energy-diffusion mean first-passage time τ_Z for large Q).","core_discovery":"The first-passage-time density of an underdamped harmonic oscillator is exactly the sum of three contributions: an equilibrium Dirac mass for initial conditions already above threshold, a Hamiltonian flux of high-energy states that empties within one oscillation period, and a long-time survival probability whose rate is the slowest eigenvalue of the absorbing Kramers operator (or the energy-diffusion mean time plus half a period when Q is very large). These expressions match numerical Langevin trajectories for all tested Q and barrier heights.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Underdamped FPT density: Dirac mass + Hamiltonian flux + Kramers eigenvalue","Three Q regimes build exact first-passage density of harmonic oscillators","Hamiltonian flux, Kramers modes and energy diffusion fix underdamped FPT","First-passage times split into equilibrium, flux and slow Kramers pieces","Underdamped oscillator FPT is sum of three Q-dependent contributions"],"cache_read_input_tokens":128,"weakest_assumption_plain":"For intermediate times shorter than one relaxation period the authors replace the true time-dependent escape rate by the long-time eigenvalue evaluated at the moving lowest-contact energy; if that educated guess fails, intermediate plateaus are no longer predicted quantitatively.","fun_headline_variants_meta":{"raw":{"variants":["Underdamped FPT density: Dirac mass + Hamiltonian flux + Kramers eigenvalue","Three Q regimes build exact first-passage density of harmonic oscillators","Hamiltonian flux, Kramers modes and energy diffusion fix underdamped FPT","First-passage times split into equilibrium, flux and slow Kramers pieces","Underdamped oscillator FPT is sum of three Q-dependent contributions"]},"model":"grok-4.5","effort":"low","cost_usd":0.003814,"raw_usage":{"total_tokens":1189,"prompt_tokens":734,"num_sources_used":0,"completion_tokens":103,"cost_in_usd_ticks":38140000,"prompt_tokens_details":{"text_tokens":734,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":352,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":734,"tokens_out":103,"duration_ms":3758,"temperature":1.0,"reasoning_tokens":352,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T08:50:50.391292+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A long Langevin trajectory (or experiment) at moderate Q whose first-passage histogram fails to match the three-piece formula, especially the height and duration of the early plateaus controlled by the instantaneous contact energy E†(t).","supporting_citations":[],"review_version":2}