Steady-state analysis of mean first-passage times with stochastic switching

One can find the mean-first passage time (MFPT) of a particle diffusing in a closed domain by solving the adjoint Smoluchowski equation with appropriate boundary conditions. An alternative approach to determining the MFPT that exploits the fact that the MFPT is given by the inverse of the Kramers flux-over-population ratio was proposed by Reimann, Schmid, and Hanggi (RSH). In this approach to find the MFPT, one has to determine the steady-state number of particles in the domain maintained by a constant flux injected at the particle starting point and divide this number by the injected flux. Here, we consider the MFPT of a particle diffusing in a cylindrical cavity to a circular spot of arbitrary radius located on the cavity base and generalize the RSH approach to the case where the spot radius and particle diffusivity stochastically switch between two values. This generalization allows one to find the MFPT as a function of the particle diffusivities and the spot radii in the two states, the cavity length and radius, the particle starting distance from the cavity base, and the switching rates. Comparison of our theoretical predictions with the MFPT obtained from Brownian dynamics simulations shows excellent agreement between the two.