First-passage time to capture for diffusion in a 3D harmonic potential

TL;DR

This paper analytically derives the first-passage time distribution for a particle diffusing in a 3D harmonic potential with an absorbing boundary, correcting previous results and validating with simulations.

Contribution

It provides a corrected analytical solution for the survival probability and first-passage time in a harmonic trap, verified by simulations and sum over eigenfunctions.

Findings

Analytical solution for survival probability as an eigenfunction sum.

Validated FPT distribution with simulations across parameters.

Derived mean first-passage time matching previous methods.

Abstract

We determine the survival probability and first-passage time (FPT) to capture for a harmonically trapped particle, diffusing outside an absorbing spherical boundary by directly solving the differential equation for the survival probability. This solution, obtained as an infinite sum over the relevant eigenfunctions, corrects previously published results [D. S. Grebenkov, J. Phys. A 48, 013001 (2014)]. To verify our calculations, we perform simulations of the survival probability, that accurately reproduce the analytic solutions for a range of parameter values. We then obtain the corresponding FPT distribution as the negative time derivative of the survival probability. Finally, we derive an expression for mean first-passage time (MFPT), also as a sum over eigenfunctions. Numerical evaluation of the first twenty-five terms in this sum closely matches the MFPT obtained by a different method in D. S. Grebenkov, J. Phys. A 48, 013001 (2014). We also find that, in the limit of vanishing trap stiffness, the amplitude of the first term in our infinite-sum solution for the survival probability matches the theoretical escape probability for the potential-free diffusion-to-capture process.