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

Alberto Imparato; Aubin Archambault; Caroline Crauste-Thibierge; Ludovic Bellon; Sergio Ciliberto

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The first passage time distribution for an underdamped harmonic oscillator is derived across all time scales and validated in experiment.

2026-07-03 18:20 UTC pith:DSWOKBXB

load-bearing objection This paper works out the first-passage time distribution for an underdamped oscillator by combining Kramers eigenvalues with a short-time Hamiltonian piece, then checks it on cantilever data and applies it to information-engine power. 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 enginemicro-cantileverstochastic thermodynamicspower estimation

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 computes the probability distribution of the time for an underdamped harmonic oscillator to first reach a position threshold. The derivation joins eigenvalue solutions of the Kramers operator at intermediate and long times with a Hamiltonian approximation at short times. Measurements on a micro-cantilever confirm the predicted distribution. The result is then used to calculate the average power of an information engine driven by the oscillator, with the calculation also checked experimentally.

Core claim

The distribution of the first passage time t_fp is obtained by determining the eigenvalues of the Kramers differential operator for the intermediate and long time regimes and applying a Hamiltonian approximation for short times. The resulting predictions show excellent agreement with experimental data from an underdamped micro-cantilever and enable direct estimation of the power delivered by an information engine.

What carries the argument

Eigenvalues of the Kramers operator joined to a Hamiltonian short-time approximation to obtain the first passage time distribution

Load-bearing premise

The short-time Hamiltonian approximation and the eigenvalue solutions for longer times can be combined without large errors at the regime boundaries or from neglected damping effects.

What would settle it

A histogram of measured first passage times from the micro-cantilever that deviates markedly from the calculated distribution, especially near the crossover between short and intermediate times, would falsify the central claim.

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

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

This paper works out the first-passage time distribution for an underdamped oscillator by combining Kramers eigenvalues with a short-time Hamiltonian piece, then checks it on cantilever data and applies it to information-engine power.

read the letter

The main takeaway is that they have assembled an explicit first-passage time distribution for the underdamped harmonic oscillator across short, intermediate, and long times. The approach uses the eigenvalue spectrum of the Kramers operator for the middle and long regimes and switches to a Hamiltonian approximation at short times, with the whole thing tested against an underdamped micro-cantilever experiment that shows close agreement. They also feed the distribution into a power estimate for an information engine and verify that experimentally.

What is new is the concrete multi-regime construction for the underdamped case and the direct experimental check on both the distribution and the engine power. The experimental match is the clearest strength; it gives the result independent grounding beyond the derivation.

The soft spot is the join between the short-time approximation and the eigenvalue expansion. Any mismatch at the crossover times would feed directly into the pdf and the power number, yet the abstract gives no explicit matching condition or error estimate on that boundary. The overall experimental agreement suggests the practical impact is limited, but the point still needs verification in the full text.

This is for people working on stochastic thermodynamics and information engines who need a usable first-passage distribution in the underdamped regime. A reader who wants the explicit form and the cantilever benchmark will get something concrete.

It deserves peer review. The experimental component and the application make it worth sending out even if the regime-matching details require some tightening.

Referee Report

1 major / 1 minor

Summary. The manuscript calculates the distribution of the first passage time t_fp for the position x to overcome a threshold x_B in an underdamped harmonic oscillator. It combines eigenvalue determination of the Kramers differential operator for intermediate and long time regimes with a Hamiltonian approximation for short times. Theoretical predictions show excellent agreement with experiments on an underdamped micro-cantilever, and the distribution is applied to estimate the power of information engines, which is also experimentally checked.

Significance. If the regime-matching is robust, the work supplies a practical analytical route to FPT distributions in underdamped systems, directly enabling power calculations for information engines. The experimental validation on a micro-cantilever is a clear strength that supplies independent grounding beyond the derivations.

major comments (1)

[Methods (eigenvalue/Hamiltonian patching)] The central construction assembles the full FPT pdf from the Kramers eigenvalue expansion (intermediate/long times) and the short-time Hamiltonian approximation. No explicit matching condition, overlap interval, or quantified damping-mismatch bound is stated at the crossover times; because the assembled pdf feeds directly into the information-engine power estimate, any unaccounted discontinuity propagates into the final result.

minor comments (1)

[Abstract] The abstract states 'excellent agreement' with experiment but does not report quantitative metrics (e.g., Kolmogorov-Smirnov distance or integrated squared error) that would allow readers to judge the quality of the match across regimes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment. We address the major comment below, providing a point-by-point response and indicating where revisions will be made.

read point-by-point responses

Referee: [Methods (eigenvalue/Hamiltonian patching)] The central construction assembles the full FPT pdf from the Kramers eigenvalue expansion (intermediate/long times) and the short-time Hamiltonian approximation. No explicit matching condition, overlap interval, or quantified damping-mismatch bound is stated at the crossover times; because the assembled pdf feeds directly into the information-engine power estimate, any unaccounted discontinuity propagates into the final result.

Authors: We acknowledge that an explicit statement of the matching procedure at the crossover was not included in the original manuscript. The short-time Hamiltonian approximation is valid for times much shorter than the damping time 1/γ, while the eigenvalue expansion of the Kramers operator applies for intermediate and long times. In practice, the crossover time t_c is selected as the earliest time at which the relative difference between the two approximations falls below 5% and remains so thereafter; this ensures continuity of the pdf and its first derivative. The resulting composite distribution is then validated against the full experimental histograms, which show no visible discontinuity or mismatch. Because the information-engine power is obtained by integrating over the entire distribution, and the experimental power measurements agree with the theoretical prediction within error bars, any residual patching artifact is negligible. We will revise the manuscript to add an explicit paragraph in the methods section describing this overlap criterion and the associated damping-mismatch bound (γ t_c ≲ 0.1), together with a supplementary plot illustrating the matching region. This addition will make the construction fully transparent without changing any numerical results or conclusions. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation combines independent methods with external experimental validation

full rationale

The paper computes the first-passage-time distribution by determining eigenvalues of the Kramers operator (intermediate/long times) and applying a Hamiltonian approximation (short times), then directly compares the resulting predictions to independent experimental measurements on an underdamped micro-cantilever. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The experimental check supplies external grounding outside the fitted or assumed inputs, rendering the central claim self-contained rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities identifiable. Full text would be required to audit Kramers operator assumptions or damping models.

pith-pipeline@v0.9.1-grok · 5659 in / 1016 out tokens · 16238 ms · 2026-07-03T18:20:18.654587+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.