Two definitions of the hopping time in a confined fluid of finite particles

We consider a fluid of hard disks diffusing in a flat long narrow channel of width approaching from above the doubled diameter of the disks. In this limit, the disks can pass their neighbors only rarely, in a mean hopping time growing to infinity, so the disks start by diffusing anomalously. We study the hopping time, which is the crucial parameter of the theory describing the subsequent transition to normal diffusion. We show that two different definitions of this quantity, based either on the mean first passage time calculated from solution of the Fick-Jacobs equation, or coming from transition state theory, are incompatible. They have different physical interpretation and also, they give different dependencies of the hopping time on the width of the channel.