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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-

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2026-07-12 08:51 UTC pith:DSWOKBXB

load-bearing objection 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. the 1 major comments →

arxiv 2607.01404 v2 pith:DSWOKBXB submitted 2026-07-01 cond-mat.stat-mech

First passage time for an underdamped harmonic oscillator and application to the power of an information engine

Aubin Archambault , Caroline Crauste-Thibierge , Alberto Imparato , Sergio Ciliberto , Ludovic Bellon This is my paper
classification cond-mat.stat-mech
keywords first passage timeunderdamped harmonic oscillatorKramers operatorinformation engineenergy diffusionmicro-cantileverstochastic thermodynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

Core claim

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.

What carries the argument

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).

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

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.

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 (1)
  1. 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.
minor comments (4)
  1. 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.
  2. 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.
  3. 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.
  4. Typographical: “APPLICA TION” and “INFORMA TION” in the section heading contain spurious spaces; “an” should be “and” in the abstract sentence on engine power.

Circularity Check

0 steps flagged

No significant circularity: FPT pdf is derived from Langevin dynamics plus equilibrium measure, then independently validated by experiment.

full rationale

The claimed closed-form pdf (Eq. 2) is assembled from three pieces that follow directly from the underdamped Langevin equation (1) and the equilibrium Boltzmann measure. P_I is the exact integral of the position Boltzmann factor above x_B (Eq. 3). P_II is obtained by integrating the same measure over the Hamiltonian phase-space region that reaches the threshold inside one period, yielding the explicit plateau formula (Eq. 12) with the kinematic cutoff E†(t) (Eqs. 11). P_III is the survival probability generated by the slowest eigenvalue of the Kramers operator (or its energy-diffusion approximation) with absorbing boundary at x_B; the time-dependent rate Γ(t)=λ1[E†(t)] is used only until t=2π and then becomes the constant Kramers rate. None of these steps defines a quantity in terms of the quantity being predicted, nor is any free parameter fitted to the FPT histograms themselves. The experimental B and Q are measured a priori from the cantilever resonance; the histograms and the information-engine power are independent data sets against which the parameter-free formulas are compared. The companion paper supplies technical details of the spectral calculation but is not invoked as a uniqueness theorem or as the sole justification of a load-bearing ansatz. Consequently the derivation chain does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim rests on the standard underdamped Langevin dynamics, the equilibrium Boltzmann measure, the Hamiltonian approximation inside one period, and the spectral properties of the Kramers operator with an absorbing boundary. No new particles or forces are postulated; the only adjustable experimental numbers are the measured barrier B, quality factor Q and the imposed dwell time τ.

free parameters (2)
  • dwell time τ after each stroke = 11 periods
    Chosen by hand as τ = 11 × 2π ≃ 5 τ_r to guarantee return to equilibrium; enters the mean power formula (Eq. 17) and therefore the location of the power maximum.
  • experimental barrier height B = 0.52–1.64 (several runs)
    Extracted from the measured threshold x_B and the independently calibrated thermal length σ; uncertainty on B is propagated as shaded bands in Fig. 1b.
axioms (4)
  • domain assumption Underdamped Langevin equation (1) with white noise of intensity 2/Q correctly describes the micro-cantilever fundamental mode.
    Standard model for a high-Q resonator in a thermal bath; invoked from the first paragraph of the Trigger Time Distribution section.
  • domain assumption For times shorter than one period the total energy E is approximately conserved, so trajectories are arcs of the Hamiltonian flow x = √(2E) cos θ.
    Used to construct the phase-space regions I and II and to obtain the closed expression for P_II (Eq. 12).
  • domain assumption The long-time escape rate is given by the slowest eigenvalue −λ_1 of the Kramers operator with absorbing boundary at x_B (or by Zwanzig’s energy-diffusion formula when Q ≫ 1).
    Standard spectral theory of the Fokker–Planck/Kramers operator; details deferred to the companion paper.
  • standard math Initial conditions are drawn from the equilibrium Boltzmann measure P(θ,E) = e^{-E}/(2π).
    Equilibrium assumption stated at the opening of the Trigger Time Distribution section and used for both P_I and the phase-space integral for P_II.

pith-pipeline@v1.1.0-grok45 · 13881 in / 2906 out tokens · 37780 ms · 2026-07-12T08:51:08.895859+00:00 · methodology

0 comments
read the original abstract

The distribution of the first passage time $t_{fp}$ for the position $x$ to overcome a threshold $x_B$ is calculated in an underdamped harmonic oscillator. The proof combines several approaches based on the determination of the eigenvalues of the Kramers differential operator for the intermediate and long time regimes and on a Hamiltonian approximation for the short times. The theoretical predictions are in excellent agreement with the results of an experiment on an underdamped micro-cantilever. The knowledge of the $t_{fp}$ distribution opens the way to several applications, among them the precise estimation of the power of information engines, which we have also experimentally checked.

Figures

Figures reproduced from arXiv: 2607.01404 by Alberto Imparato, Aubin Archambault, Caroline Crauste-Thibierge, Ludovic Bellon, Sergio Ciliberto.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Theoretical probability distribution function [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) In the Hamiltonian dynamics approximation, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Main contributions to the pdf of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Mean power of the information engine versus [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. First passage time distribution in underdamped harmonic oscillators

    cond-mat.stat-mech 2026-07 accept novelty 7.0

    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.

  2. First passage time distribution in underdamped harmonic oscillators

    cond-mat.stat-mech 2026-07 unverdicted novelty 5.0

    Distribution of first passage times for underdamped harmonic oscillators is derived via regime-specific approximations and validated against numerical simulations.

Reference graph

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    Note that to avoid curve overlapping,P I and Γ B have been offset by a factor 2 and 1 2 respectively, as noted in the legend