Reconstruction of potential energy profiles from multiple rupture time distributions

We explore the mathematical and numerical aspects of reconstructing a potential energy profile of a molecular bond from its rupture time distribution. While reliable reconstruction of gross attributes, such as the height and the width of an energy barrier, can be easily extracted from a single first passage time (FPT) distribution, the reconstruction of finer structure is ill-conditioned. More careful analysis shows the existence of optimal bond potential amplitudes (represented by an effective Peclet number) and initial bond configurations that yield the most efficient numerical reconstruction of simple potentials. Furthermore, we show that reconstruction of more complex potentials containing multiple minima can be achieved by simultaneously using two or more measured FPT distributions, obtained under different physical conditions. For example, by changing the effective potential energy surface by known amounts, additional measured FPT distributions improve the reconstruction. We demonstrate the possibility of reconstructing potentials with multiple minima, motivate heuristic rules-of-thumb for optimizing the reconstruction, and discuss further applications and extensions.

💡 Research Summary

The paper addresses the inverse problem of reconstructing a molecular bond’s potential‑energy profile V(x) from measured rupture‑time statistics, specifically the first‑passage‑time (FPT) distribution p(t). Starting from the one‑dimensional drift‑diffusion equation that governs the stochastic motion of the bond coordinate under an external force and thermal noise, the authors derive the forward mapping V(x) → p(t) for a given initial bond length x₀ and boundary conditions (absorbing at the rupture point, reflecting elsewhere). While the forward problem is straightforward to solve numerically or experimentally, the inverse problem is severely ill‑conditioned: fine features of V(x) (high‑frequency components) leave only faint imprints in the tail of p(t) and are easily obscured by measurement noise.

To quantify and mitigate this ill‑conditioning, the study first explores how two controllable experimental parameters influence the information content of a single FPT distribution. The dimensionless Peclet number Pe = (F·L)/(k_BT) measures the relative strength of deterministic drift to stochastic diffusion. When Pe is too small, diffusion dominates and the rupture times are broadly distributed, washing out details of the barrier. When Pe is too large, drift overwhelms diffusion and the particle rushes over the barrier, again suppressing sensitivity to subtle variations in V(x). Numerical sweeps reveal an intermediate “optimal” Pe (≈5–10 for the test potentials) at which the Fisher information about the potential is maximized. A second lever is the initial bond configuration x₀. Starting close to the barrier yields rapid rupture dominated by low‑frequency information, whereas starting deep in a well produces long waiting times that emphasize low‑frequency components. By sampling p(t) at several distinct x₀ values, the effective bandwidth of the data can be broadened, improving reconstruction stability.

Recognizing that a single p(t) cannot reliably resolve complex landscapes, the authors propose a multi‑distribution strategy. By deliberately altering the external conditions—e.g., adding a known bias force ΔF, changing temperature, or applying a calibrated potential offset ΔV(x)—one generates a family of modified potentials V_i(x)=V(x)+ΔV_i(x) and corresponding FPT distributions p_i(t). These distributions are statistically independent and together provide a set of linear‑independent constraints on V(x). The inverse problem is then cast as a regularized nonlinear least‑squares optimization: \