Aubin Archambault, Caroline Crauste-Thibierge, Alberto Imparato, Sergio Ciliberto, Ludovic Bellon, submitted to Phys. Rev. E
arXiv: 2607.01405
We derive the distribution of the first passage time tfp for the position x of an underdamped harmonic oscillator to overcome a threshold xB. As the tfp distribution depends on the oscillator quality factor Q different approaches are used. At very large quality factor (Qâ«100) and intermediate and long tfp the proof is based on an energy diffusion model, whereas at medium quality factor (QâŒ10) the proof is based on the study of the eigenvalues of the Kramers linear differential operator with absorbing boundary conditions. For all Q and short tfp we use a Hamiltonian approximation. The theoretical predictions are in excellent agreement with direct numerical simulations of underdamped oscillator dynamics. Finally we show that the mean of the trajectories ending at tfp presents a particular shape driven by a specific noise pattern.