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

Technical Summary: First Passage Time for an Underdamped Harmonic Oscillator and Application to Information Engines

Problem Statement
The estimation of the First Passage Time (FPT) distribution is a fundamental problem in statistical physics with applications ranging from chemical reaction rates to information engines. While FPT distributions are well-established for overdamped systems described by Langevin equations, the underdamped regime remains challenging. In underdamped systems, inertia introduces a second derivative of displacement in the stochastic equations of motion, rendering the process non-Markovian. Consequently, computing the FPT probability has historically been a "daunting task," with closed-form solutions generally available only for specific limits (e.g., the free particle in the long-time limit). This paper addresses the calculation of the FPT distribution, $P(t_{fp})$, for a Brownian harmonic oscillator with a finite quality factor $Q$ crossing a position threshold $x_B$.

Methodology
The authors propose a unified theoretical framework that combines three distinct approaches to cover the full range of time scales and quality factors:

1. Decomposition of the PDF: The total probability density function $P(t_{fp})$ is decomposed into three contributions:

   • $P_I \delta(t_{fp})$: Represents instantaneous triggering where the initial position $x(0) \geq x_B$.
   • $P_{II}(t_{fp})$: Describes the dynamics within the first oscillation period ($0 < t_{fp} \leq 2\pi$).
   • $P_{III}(t_{fp})$: Captures the long-time behavior ($t_{fp} > 2\pi$).

2. Short-Time Regime ($t_{fp} \leq 2\pi$): For low damping ($Q \gg 1$), the authors employ a Hamiltonian approximation. They assume the total energy $E$ is approximately constant over one period.

   • Instantaneous Trigger ($P_I$): Calculated using the Boltzmann distribution for the initial position, yielding $P_I = \frac{1}{2}\text{erfc}(\sqrt{B})$, where $B = \frac{1}{2}x_B^2$ is the energy barrier.
   • First Period ($P_{II}$): The dynamics are treated as deterministic sinusoidal motion $x(t) = x_E \cos(\theta_0 + t)$. The probability is derived by integrating over the joint distribution of initial energy $E$ and phase $\theta_0$. This yields a plateau followed by a rapid decay, determined by the maximum energy $E^\dagger(t)$ accessible at time $t$.

3. Long-Time Regime ($t_{fp} > 2\pi$): For times exceeding one oscillation period, the system's behavior is governed by energy diffusion due to thermal noise.

   • High $Q$ Limit: The problem is mapped to a Kramers escape problem across an energy barrier. The escape rate $\Gamma(t)$ is computed based on the time-dependent energy threshold $E^\dagger(t)$, eventually converging to a constant rate $\Gamma_B$.
   • Finite $Q$: The authors determine the escape rate $\Gamma$ by calculating the slowest eigenvalue $-\lambda_1(x_B)$ of the Kramers differential operator associated with the stochastic equation, subject to an absorbing boundary condition at $x_B$.
   • The resulting tail $P_{III}$ follows an exponential decay $e^{-\Gamma_B t_{fp}}$.

Key Results
The paper presents a closed-form analytical expression for the FPT distribution (Eq. 2) parameterized solely by the energy barrier $B$ and the quality factor $Q$.

• Experimental Validation: The theoretical predictions were tested against an experiment using an underdamped micro-cantilever ($Q \approx 7$) at room temperature. The position was measured with picometer resolution at a 2 MHz sampling rate.
• Agreement: The experimental data showed "excellent agreement" with the theoretical model across three distinct regimes:
  1. The probability of instantaneous triggers ($P_I$).
  2. The plateau value of the distribution for $0 < t < \pi$ ($P_{II}$).
  3. The exponential decay rate of the long-time tail ($P_{III}$).
• Information Engine Application: The derived FPT distribution was applied to estimate the power of an information engine. In this setup, work is extracted when the oscillator crosses a threshold $x_B$, shifting the potential well. The mean power $\langle P \rangle$ was calculated as the ratio of mean extracted work to the sum of the mean FPT and a dwell time. The theoretical power curve matched experimental data, identifying an optimal operating point where the threshold $x_B \approx 2$ and the shift $L \approx 1$.

Significance and Claims
The authors claim that this work provides a "new insight" into the FPT problem for underdamped systems by offering a solution that covers the full range of quality factors, bridging the gap between short-time Hamiltonian dynamics and long-time diffusive escape.

• Theoretical Advancement: The paper asserts that the derived closed-form solution (Eq. 2) represents an advancement in FPT statistics for underdamped systems. It notes that previously, the only known closed-form solution for an underdamped system was for the free particle (random acceleration process) in the long-time limit.
• Distinction from Kramers Theory: The authors emphasize that their description differs from standard Kramers rate theory, which typically applies to diffusive crossing of large barriers. In their distribution, the exponential tail only emerges for times significantly larger than one oscillation period, while the short-time behavior is dominated by inertial oscillations.
• Practical Utility: The method enables the precise estimation of power in information engines with inertia, a parameter previously difficult to quantify without full distribution knowledge. The authors state that this opens the way for applying FPT statistics to other fields involving underdamped dynamics, such as stochastic thermodynamics.

The paper concludes by noting that the full details of the calculations and numerical simulations regarding the effect of the quality factor are provided in an accompanying paper [22].