Brownian dynamics mean first passage time of two hard disks diffusing in a channel

We use Brownian dynamics simulations of two hard disks in a channel to study the mean first passage time to pass each other. The disks have a diameter sigma and are confined in a channel with hard reflective walls. The mean first passage time diverges with an exponent eta as the channel width (2R(p)) approaches that of the nonpassing limit (2sigma). There are two different theoretical predictions for the exponent eta of the two disk hopping time divergences. Transition state theory and a Fick-Jacobs type of dimensional reduction approach predict exponents of 2 and 32, respectively. Previous Brownian dynamics simulations results have a range of effective exponents and are inconclusive. Here, we present extensive Brownian dynamics simulations results which are consistent with the predictions of transition state theory. The new data show that one must be close to the nonpassing limit to observe the asymptotic scaling exponent. The scaling dependence crosses over from the bulk limit to the nonpassing limit as the width of the channel narrows, corresponding to a range of effective exponents between 0 and 2. This crossover provides an explanation of the inconclusive results reported in previous Brownian dynamics simulations.