Diffusion Limited Escape Rate of a Complex Molecule in Multi-dimensional Confinement

The rate at which a Brownian particle confined in a closed space escapes from the space by passing through a narrow passage is called the escape rate. The escape rate is relevant to many diffusion limited processes in polymer and colloidal systems, such as colloidal aggregation, polymerization reaction, polymer translocation through a membrane, etc. Here, we propose a variational principle to calculate the escape rate of complex molecules doing Brownian motion in a multi-dimensional phase space. We propose a regional minimization method in which we divide the whole phase space into regions, conduct the minimization for each region, and combine the results to get the minimum in the entire space. As an example, we discuss (1) the escape rate of a point particle that escapes from a confinement passing through a long corridor and (2) the escape rate of a rod-like particle that escapes through a small hole made in the wall of the confinement.