The influence of absorbing boundary conditions on the transition path times statistics

TL;DR

This paper derives an analytical expression for the transition path time distribution of a particle crossing a parabolic barrier, considering absorbing boundary conditions and anomalous dynamics, with validation against simulations and implications for experimental analysis.

Contribution

It extends existing TPT distribution models by incorporating absorbing boundary conditions and anomalous dynamics, providing a more accurate framework for experimental data analysis.

Findings

Analytical TPT distribution derived for parabolic barriers with absorbing boundaries.

Numerical simulations agree well with the analytical results.

Free boundary conditions overestimate barrier height in TPT analysis.

Abstract

We derive an analytical expression for the transition path time (TPT) distribution for a one-dimensional particle crossing a parabolic barrier. The solution is expressed in terms of the eigenfunctions and eigenvalues of the associated Fokker-Planck equation. The particle performs an anomalous dynamics generated by a power-law memory kernel, which includes memoryless Markovian dynamics as a limiting case. Our result takes into account absorbing boundary conditions, extending existing results obtained for free boundaries. We show that TPT distributions obtained from numerical simulations are in excellent agreement with analytical results, while the typically employed free boundary conditions lead to a systematic overestimation of the barrier height. These findings may be useful in the analysis of experimental results on transition path times. A web tool to perform this analysis is freely available.