Two-Dimensional Active Brownian Particles Crossing a Parabolic Barrier: Finite Rectangular Domain with Absorbing Boundary Conditions

TL;DR

This paper analytically solves the Fokker-Planck equation for a 2D active Brownian particle in a rectangular domain with a parabolic barrier, revealing how activity affects survival and first-passage times.

Contribution

It introduces a matrix-based perturbative method to solve the time-dependent Fokker-Planck equation for active particles in complex geometries with absorbing boundaries.

Findings

Activity significantly alters survival probabilities and first-passage times.

Rotational diffusivity has a minor effect on these quantities.

The approach provides explicit expressions for propagators and eigenvalues.

Abstract

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a rectangular domain with absorbing boundary and in the presence of a parabolic barrier along one direction. By taking those of a passive Brownian particle as basis states and dealing with the activity as a perturbation, we provide a matrix representation of the Fokker-Planck operator and express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Our solution also allows us to obtain the survival probability and the first-passage-time distribution. The non-equilibrium character of the dynamics induces a strong dependence of the latter quantities on the particle's activity, while the rotational diffusivity influences them to a minor extent.