Mean first passage time for a small rotating trap inside a reflective disk

Abstract: We compute the mean first passage time (MFPT) for a Brownian particle inside a two-dimensional disk with reflective boundaries and a small interior trap that is rotating at a constant angular velocity. The inherent symmetry of the problem allows for a detailed analytic study of the situation. For a given angular velocity, we determine the optimal radius of rotation that minimizes the average MFPT over the disk. Several distinct regimes are observed, depending on the ratio between the angular velocity $\omega$ and the trap size $\varepsilon$, and several intricate transitions are analyzed using the tools of asymptotic analysis and Fourier series. For $\omega \sim \mathcal{O}(1)$, we compute a critical value $\omega_c>0$ such that the optimal trap location is at the origin whenever $\omega < \omega_c$, and is off the origin for $\omega>\omega_c $. In the regime $1 \ll \omega \ll \mathcal{O}(\varepsilon^{-1})$ the optimal trap path approaches the boundary of the disk. However as $\omega$ is further increased to $\mathcal{O}(\varepsilon^{-1})$, the optimal trap path "jumps" closer to the origin. Finally for $\omega \gg \mathcal{O}(\varepsilon^{-1})$ the optimal trap path subdivides the disk into two regions of equal area. This simple geometry provides a good test case for future studies of MFPT with more complex trap motion.