Spatiotemporal Characterization of Active Brownian Dynamics in Channels

Abstract

Accumulation at boundaries represents a widely observed phenomenon in active systems with implications for microbial ecology and engineering applications. To rationalize the underlying physics, we provide analytical predictions for the first-passage properties and spatial distributions of a confined active Brownian particle (ABP). We show that ABPs with absorbing and hard-wall boundary conditions are Siegmund duals, yielding a direct mapping between the propagators of the two problems. We analyze the system across low and high activity regimes -- quantifying persistent motion relative to diffusion -- and show that active motion, together with a favorable initial orientation, typically lowers the mean first-passage time relative to passive diffusion. Notably, the full time-dependent propagator between hard walls approaches a wall-accumulated stationary state given by the derivative of the splitting probability as a consequence of Siegmund duality.

Figures (3)

• Figure 1: Schematic of the duals: ABPs between absorbing (sticking) and hard (reflective) walls. The left side shows an ABP hitting an absorbing boundary, which gives access to first-passage properties. The right side illustrates the dynamics between hard walls, yielding particle distributions. The inset depicts the agent's position $\boldsymbol{r}$ and orientation $\vartheta$.
• Figure 2: First-passage properties. (a) Mean-first-passage time $\langle T \rangle$ normalized by its maximum $T_{\mathrm{max}}$, as a function of the initial position $z_{0}$. (b) Mean-first-passage time $\langle T \rangle$ as a function of the Péclet number and for $z_{0}/L=0.2$. (c) Splitting probability $\pi_{L}$ at the right wall as a function of $z_0$. In (a--c) we set $\gamma=3$. In (b), dashed and solid lines denote the low-$\mathrm{Pe}$ expansion up to $O(\mathrm{Pe}^2)$ and the high-activity result [Eq. \ref{['eq:mfpt_high_pe']}], respectively. Symbols correspond to simulation results.
• Figure 3: Probability densities of an ABP between hard walls. (a) Probability density $p_{H}(z,t|z_0)$ as a function of the position $z$ for different times $t$. Here, $z_{0}/L =0.8$ and $\mathrm{Pe}=0.5$. Lines correspond to the low-$\mathrm{Pe}$ (up to $O(\mathrm{Pe}^{2})$) expansion and symbols are simulation results. (Inset) Probability density $p_{H}(z,t|z_0)$ for the case of a single wall at $z=0$. We set $z_{0}/a =4$ and $\mathrm{Pe}=0.5$, and rescale length scales with the particle's hydrodynamic radius $a = \sqrt{3D/(4D_{\mathrm{rot}})}$. (b) Stationary distribution $p_{H}(z)$ for different Péclet numbers. Lines correspond to the high-$\rm Pe$ solution \ref{['eq:pH_stationary']} and symbols are simulation results. (a--b) Vertical gray lines indicate the walls.