Two-Dimensional Active Brownian Particles Crossing a Parabolic Barrier: Transition-Path Times, Survival Probability, and First-Passage Time

TL;DR

This paper derives analytical expressions for the transition path time distribution, survival probability, and first-passage times of a two-dimensional active Brownian particle crossing a parabolic barrier, highlighting the impact of activity on these quantities.

Contribution

It introduces a perturbative analytical solution for active Brownian particles crossing barriers, extending passive particle models by incorporating activity effects.

Findings

Self-propulsion shortens transition path times.

Activity significantly influences survival probability and first-passage times.

Rotational diffusivity has a minor effect on these quantities.

Abstract

We derive an analytical expression for the propagator and the transition path time distribution of a two-dimensional active Brownian particle crossing a parabolic barrier with absorbing boundary conditions at both sides. By taking those of a passive Brownian particle as basis states and dealing with the activity as a perturbation, our solution is expressed in terms of the perturbed eigenfunctions and eigenvalues of the associated Fokker-Planck equation once the latter is reduced by taking into account only the coordinate along the direction of the barrier and the self-propulsion angle. We show that transition path times are typically shortened by the self-propulsion of the particle. Our solution also allows us to obtain the survival probability and the first-passage times distribution, which display a strong dependence on the particle's activity, while the rotational diffusivity influences them to a minor extent.