Splitting probabilities of confined active particles

Abstract

Active particles exhibit self-propulsion, leading to transport behavior that differs fundamentally from passive Brownian motion. In confined or structured domains, activity strongly influence escape probabilities and first-passage behavior. Understanding these effects is essential for describing transport in biological microenvironments, microfluidic devices, and heterogeneous media. In this work, leveraging the backward Fokker--Planck equation, we investigate the splitting probability of active particles in confined domains, focusing on both a one-dimensional interval and a two-dimensional corrugated channel. Analytical solutions are derived for the one-dimensional case in various asymptotic regimes. In corrugated channels with small aspect ratios, we develop a Fick--Jacobs reduction that yields effective transport equations along the axial direction, whereas for finite aspect ratios, the splitting dynamics are characterized numerically. We demonstrate how channel geometry, particle activity, and chirality modulate the likelihood of escape through different boundaries. Our results provide quantitative predictions for the transport of active matter in complex environments and highlight the interplay between confinement and activity.

Figures (8)

• Figure 1: Schematic illustration of an active particle in a one-dimensional interval. The orientation vector, $\bm{q}$, forms an angle $\theta$ with the horizontal ($x$) axis.
• Figure 2: Plots of $C_0(x)$, defined in Eq. \ref{['Eq:C0']}, as a function of the initial position of the particle $x$ for different values of chirality ($\chi$). In all cases, $\gamma=0.1$.
• Figure 3: Plots of the splitting probability $\langle p_R\rangle$ as a function of the particle’s initial position $x$. (a) $\langle p_R\rangle$ for ABPs ($\chi=0$) at different values of $Pe$. (b) $\langle p_R\rangle$ for CAPs at different values of chirality $\chi$ ($Pe=10$). In all cases, $\gamma=0.1$.
• Figure 4: (a) Comparison between the numerical (FEM) and leading-order asymptotic solutions for $\langle p_R \rangle$ at $Pe=10$. We note that the asymptotic solution for $\langle p_R \rangle$ is independent of both $\gamma$ and $\chi$. (b) Comparison between the numerical (FEM) and leading-order asymptotic solutions for $p_R(x,\pi)$ at $Pe=10$. We note that in panel (b), the numerical and asymptotic solutions are visually indistinguishable. (c) Contour plot of the numerical solutions for $p_R$ at $Pe=10$. For the numerical results shown, we used $Pe=10$, $\gamma=0.1$, and $\chi=0$.
• Figure 5: Schematic illustration (not to scale) of an active particle in the unit cell of a periodic corrugated channel. In the theoretical formulation, the particle is treated as a point. The dashed lines indicate the left and right exits.