Mean first passage time of active Brownian particles in two dimensions

TL;DR

This work derives an elliptic PDE for the mean first passage time (MFPT) of active Brownian particles (ABPs) in 2D domains, given by $U_s \bm{q}\cdot\nabla T + D_x \nabla^2 T + D_R \nabla_R^2 T = -1$, with appropriate absorbing and reflecting boundary conditions. It analyzes ABPs in disk, annulus, and ellipse geometries, and develops a weak-activity asymptotic expansion $T = T_0 + Pe_s T_1 + O(Pe_s^2)$ where $T_0$ is the passive MFPT and $T_1$ captures orientation-dependent corrections, e.g., in the disk $T_1(r,\phi)=A_1(r)\cos\phi$ with $A_1(r)=-\frac{1}{2\beta}\left( r-\frac{I_1(\sqrt{\beta}\,r)}{I_1(\sqrt{\beta})} \right)$. The results reveal non-monotonic MFPT with respect to initial position and orientation, and show that increasing swim speed can either increase or decrease MFPT depending on geometry and orientation; in some annular configurations there is an optimal swim speed that maximizes MFPT. Numerical PDE solutions are validated against Monte Carlo simulations, highlighting the interplay between activity, geometry, and boundary conditions. The findings have implications for design and control of microswimmer transport in confined environments and point to natural extensions to 3D and more complex domains.

Abstract

The mean first passage time (MFPT) is a key metric for understanding transport, search, and escape processes in stochastic systems. While well characterized for passive Brownian particles, its behavior in active systems-such as active Brownian particles (ABPs)-remains less understood due to their self-propelled, nonequilibrium dynamics. In this paper, we formulate and analyze an elliptic partial differential equation (PDE) to characterize the MFPT of ABPs in two-dimensional domains, including circular, annular, and elliptical regions. For annular regions, we analyze the MFPT of ABPs under various boundary conditions. Our results reveal rich behaviors in the MFPT of ABPs that differ fundamentally from those of passive particles. Notably, the MFPT exhibits non-monotonic dependence on the initial position and orientation of the particle, with maxima often occurring away from the domain center. We also find that increasing swimming speed can either increase or decrease the MFPT depending on the geometry and initial orientation. Asymptotic analysis of the PDE in the weak-activity regime provides insight into how activity modifies escape statistics of the particles in different geometries. Finally, our numerical solutions of the PDE are validated against Monte Carlo simulations.

Figures (10)

• Figure 1: MFPT in a disk. (a) Schematic diagram of an active Brownian particle in a disk. The radial position and angle of the particle is represented by $r$ and $\theta_1$, respectively, and its orientation angle is $\theta_2$. The angle between the radial direction and the orientation vector ($\bm{q}$) is denoted by $\phi$. (b) Contour plot of the MFPT for the particle to escape the disk for $Pe_s=5$ and $\beta=0.1$. (c) MFPT as a function of the starting radial position $r$ for $\phi = \pi$, where the particle initially points toward the origin, computed using the PDE in (\ref{['Eq:Dimless_ABPS_PDE_Polar']}) and Monte Carlo (MC) simulations.
• Figure 2: Effect of the $\mathcal{O}(Pe_s)$ term $T_1(r, \phi)$ (Eq. (\ref{['eq:T1-sol']})) on the MFPT. (a) Plots of the radial component, $A_1(r)$, as a function of the initial radial position ($r$), for different values of the dimensionless rotational diffusivity ($\beta$): $\beta=0.1$ (red curve) and $\beta=1$ (blue curve). (b) Contour plot of $T_1(r, \phi)$ with respect to $r$ and the initial orientation ($\phi$) for $\beta=1$.
• Figure 3: Plots of the radial component $A_2(r)$ of the $\mathcal{O}(Pe_s)$ term in the MFPT expansion for an annular region, as a function of the initial radial position ($r$), for different values of the dimensionless rotational diffusivity ($\beta$): $\beta=0.1$ (red curve) and $\beta=1$ (blue curve). Both the inner boundary (with radius $\alpha=0.2$) and outer boundary at $r=1$ are absorbing.
• Figure 4: MFPT in an annulus with absorbing inner and outer boundaries for $\alpha=0.2$ and $\beta=0.1$. Contour plots (a--c) show numerical solutions of the PDE in Eq. (\ref{['Eq:Dimless_ABPS_PDE_Polar']}) for different values of $Pe_s$. (d) MFPT as a function of the Péclet number $(Pe_s)$ for $r=0.8$, $\phi=0$ (blue) and $\phi=\pi$ (red). For $\phi=\pi$, there is an optimum swim speed ($Pe_s$) that maximizes the MFPT.
• Figure 5: MFPT in an annulus as a function of the starting radial position for different values of $Pe_s$. Both the inner and outer boundaries are absorbing. (a) MFPT for $\phi=\pi$. (b) MFPT for $\phi=0$. The other parameters are fixed at $\beta=0.1$ and $\alpha=0.2$. The line plots correspond to data at horizontal slices of the contour plots in Fig. \ref{['fig:annulus-bc0']}(a--c).