Active Brownian Particles in a Circular Disk with an Absorbing Boundary

TL;DR

This paper analytically solves the Fokker-Planck equation for active Brownian particles in a circular domain with absorbing boundaries, revealing how activity influences survival and first-passage times.

Contribution

It introduces a matrix approach using passive particle basis states and perturbation theory to solve the time-dependent Fokker-Planck equation for active particles.

Findings

Survival probability depends strongly on particle activity.

First-passage time distribution shows non-equilibrium effects.

Eigenfunction expansion captures activity-induced dynamics.

Abstract

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a circular region with an absorbing boundary. Using the 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 we express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Alternatively, we show that the propagator can be expressed as a combination of the equilibrium eigenstates with weights depending only on time and on the initial conditions, and obeying exact iterative relations. Our solution allows also obtaining the survival probability and the first-passage time distribution. These latter quantities exhibit peculiarities induced by the non-equilibrium character of the dynamics, in particular, they display a strong dependence on the activity of the particle and, to a less extent, also on its rotational diffusivity.