Active Brownian particles in a circular disk with an absorbing boundary

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle explor-ing 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 nonequilibrium 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.

Automated summary

In this paper, we solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a circular region with an absorbing boundary. We provide a matrix representation of the Fokker-Planck operator using the passive Brownian particle as basis states and treating the activity as a perturbation. The propagator is expressed in terms of perturbed eigenvalues and eigenfunctions or as a combination of equilibrium eigenstates with weights dependent on time and initial conditions, following exact iterative relations. Additionally, we obtain the survival probability and the first-passage time distribution, which exhibit peculiarities induced by the nonequilibrium nature of the dynamics, such as strong dependence on the particle's activity and, to a lesser extent, its rotational diffusivity.