Brownian particle escaping through a small window in a circular domain
The MFPT for a Brownian particle in a disk of radius R to escape through a window of arc-length ε is:
T ≈ (πR²)/(Dε) · (R/π) ln(R/ε) + O(1)
The logarithmic divergence as ε→0 is a hallmark of 2D diffusion. This has applications in cell biology (receptor binding, ion channels).