Particle dynamics in biconical cavities: First-passage, direct-transit, and looping time distributions

Earlier, we analyzed the effects of monotonically changing entropy potentials imposed by expanding or narrowing tubes on particle diffusion in such tubes [Berezhkovskii et al., J. Chem. Phys. 147, 134104 (2017)]. In the present study, we examine particle dynamics in biconical cavities, wherein particle motion is influenced by either an entropy potential well, as in a cavity composed of first expanding and then narrowing cones, or an entropy potential barrier, as in a cavity made up of first narrowing and then expanding identical cones. Both types of cavities are relevant to multiple technological and biological problems, where examples of such structures can be found at the micro- and nanoscales. We derive analytical expressions for the Laplace transforms of the distributions for the first-passage, direct-transit, and looping times in such structures. We find that not only the average values but also the distributions of the first-passage times in both cavities are indeed identical. However, the direct-transit and looping time distributions are drastically different. In particular, the mean direct-transit time for the expanding-narrowing cavity (entropy potential well) approaches a constant value with the increasing ratio of the cavity's largest radius to the radius of its opening. In contrast, it goes to infinity in the case of the narrowing-expanding cavity (entropy potential barrier).