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

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.

💡 Research Summary

In this paper the authors present an exact analytical treatment of a two‑dimensional active Brownian particle (ABP) confined to a finite rectangular domain with perfectly absorbing walls on all sides, while a parabolic potential U(x)=−k x²/2 acts along the x‑direction. The problem is tackled by regarding the activity (self‑propulsion speed v) as a perturbation of the well‑known passive Brownian particle (v = 0) solution. First, the passive Fokker‑Planck operator L₀ is diagonalised. By separating variables the eigenfunctions are written as a product of a complex exponential in the orientation angle θ, a sine mode in the transverse coordinate y (enforced by the absorbing boundaries at y = 0 and y = 2d_y), and a one‑dimensional x‑dependent function Y_n(x) that solves a Sturm‑Liouville problem with a parabolic drift. Y_n(x) is expressed through Kummer’s confluent hypergeometric function 1F1, and the absorbing conditions at x = ±d_x quantise the parameter σ_n. The corresponding eigenvalues are
λ_{n,m,s}=βk d_x² σ_n + (mπ/2α)² + γ s²,
where m labels the y‑mode, s the angular Fourier mode, α = d_y/d_x the aspect ratio, and γ = D_rot τ the dimensionless rotational diffusivity (τ = d_x²/D is the diffusive time across the domain).

The activity enters through the perturbation operator L₁ = −(d/τ)