First passage time distribution in underdamped harmonic oscillators

Technical Summary: First Passage Time Distribution in Underdamped Harmonic Oscillators

Problem Statement
The paper addresses the calculation of the First Passage Time distribution (FPTd), denoted $P(t_{fp})$, for the position $x$ of an underdamped harmonic oscillator to overcome a threshold $x_B$. While FPTd is well-established for overdamped systems, it remains complex for underdamped systems (characterized by a high quality factor $Q$) due to inertia-induced long-memory effects. The distribution's shape is strongly dependent on $Q$, necessitating different theoretical approaches for different damping regimes. The authors aim to provide a comprehensive derivation of $P(t_{fp})$ covering the full range of $Q$, from intermediate values ($Q \sim 10$) to very large values ($Q \gg 100$), and to analyze the specific noise patterns driving escape events.

Methodology
The authors model the system using the Langevin equation for a harmonic oscillator with mass $m$, stiffness $k$, and viscous damping $\gamma$, coupled to a thermal bath at temperature $T$. The dynamics are analyzed in the phase space $(x, v)$ using dimensionless units. The derivation of $P(t_{fp})$ is segmented into three distinct contributions based on time scales and initial conditions:

1. Instantaneous Contribution ($P_I$): Accounts for trajectories starting above the threshold ($x(0) \ge x_B$).
2. Short-time Contribution ($P_{II}$): Accounts for trajectories with high initial energy ($E(0) \ge B$) that cross the threshold within the first oscillation period.
3. Long-time Contribution ($P_{III}$): Accounts for trajectories starting with low energy ($E(0) < B$) that require thermal activation to cross the barrier.

To derive the long-time behavior, the authors employ two distinct formalisms depending on the quality factor:

• For Intermediate $Q$ ($Q \sim 10$): The authors utilize the Kramers linear differential operator with absorbing boundary conditions. They solve for the eigenvalues and eigenfunctions of this non-self-adjoint operator. The long-time FPTd is dominated by the slowest decaying eigenvalue ($\lambda_1$), which corresponds to the escape rate.
• For Very Large $Q$ ($Q \gg 100$): An energy diffusion model is derived. By transforming to polar coordinates (energy $E$ and phase $\theta$) and averaging over the fast phase variable, the system is reduced to a Fokker-Planck equation for energy alone. The mean first passage time is calculated by solving the adjoint equation for the energy diffusion process, followed by an averaging over the initial equilibrium distribution.

Additionally, the authors analyze the mean trajectories of position, velocity, and thermal noise leading up to $t_{fp}$ using instanton theory and numerical simulations to identify specific noise patterns.

Key Results

1. Decomposition of FPTd: The total distribution is expressed as the sum $P(t_{fp}) = P_I \delta(t_{fp}) + P_{II}(t_{fp}) + P_{III}(t_{fp})$.
   • $P_I$ and $P_{II}$ are derived from Hamiltonian dynamics and are independent of $Q$.
   • $P_{III}$ depends on $Q$ and describes the escape from the potential well.
2. Regime-Dependent Escape Rates:
   • At intermediate $Q$, the escape rate $\Gamma_B$ is determined by the dominant eigenvalue $\lambda_1$ of the Kramers operator. Numerical diagonalization of this operator shows excellent agreement with direct Langevin simulations.
   • At very large $Q$, the energy diffusion approach yields a mean first passage time $\tau_Z(B)$ and a corresponding rate $\Gamma_Z$. The authors find that for large $Q$, the time to reach the threshold after crossing the energy barrier is approximately $\pi$ (half a period), leading to a total rate $\Gamma_Z = 1/(\pi + \tau_Z)$.
   • The paper notes that for intermediate $Q$, the energy diffusion approximation underestimates the mean FPT because the oscillator can cross the energy barrier multiple times before reaching the spatial threshold $x_B$.
3. Transient Behavior: For large $Q$, the FPTd exhibits a "stair-like" decay before reaching the asymptotic exponential regime. These plateaus, with a periodicity of $2\pi$, arise from the contribution of higher-order eigenvalues ($\lambda_{n>1}$) which possess imaginary parts related to the oscillation frequency.
4. Noise Patterns and Mean Trajectories: The analysis of mean trajectories ending at $t_{fp}$ reveals a specific shape: an exponential growth of a sinusoidal signal followed by exponential decay. The mean noise $\langle \eta(t) \rangle$ exhibits a distinct pattern resonant with the oscillator's frequency. The velocity distribution at the moment of crossing is found to be $P(v) = v e^{-v^2/2}$, independent of $Q$.

Significance and Claims
The authors claim that their theoretical framework provides a complete description of the FPTd for underdamped harmonic oscillators across all quality factors. The primary significance lies in:

• Unifying Approaches: Demonstrating how different mathematical tools (Kramers operator eigenvalues vs. energy diffusion) are required to accurately describe the system depending on the damping regime.
• Quantitative Agreement: The theoretical predictions show excellent agreement with both direct numerical simulations of the Langevin equation and experimental data previously measured on an oscillator with $Q=7$ (referenced in an accompanying Letter [17]).
• Physical Insight: The identification of specific noise patterns driving escape events suggests that the mean FPT in underdamped systems is not merely a statistical average but is driven by a reproducible dynamical path (instanton) characterized by a specific noise signature.

The paper concludes that while the exponential decay of the FPTd is well-understood, the transient behavior (plateaus) and the specific noise patterns driving the escape remain areas where analytical determination of higher-order effects is challenging, though their existence is clearly supported by simulation.