Kinetics of escape through a small hole

We study the time dependence of the survival probability of a Brownian particle that escapes from a cavity through a round hole. When the hole is small the escape is controlled by an entropy barrier and the survival probability decays as a single exponential. We argue that the rate constant is given by $\text{k=4Da/V,}$ where a and V are the hole radius and the cavity volume and D is the diffusion constant of the particle. Brownian dynamics simulations for spherical and cubic cavities confirmed both the exponential decay of the survival probability and the expression for the rate constant for sufficiently small values of a.