  We investigate the dynamics of two interacting diffusing particles in an
infinite effectively one dimensional system; the particles interact through a
step-like potential of width b and height phi_0 and are allowed to pass one
another. By solving the corresponding 2+1-variate Fokker-Planck equation an
exact result for the two particle conditional probability density function
(PDF) is obtained for arbitrary initial particle positions. From the
two-particle PDF we obtain the overtake probability, i.e. the probability that
the two particles has exchanged positions at time t compared to the initial
configuration. In addition, we calculate the trapping probability, i.e. the
probability that the two particles are trapped close to each other (within the
barrier width b) at time t, which is mainly of interest for an attractive
potential, phi_0<0. We also investigate the tagged particle PDF, relevant for
describing the dynamics of one particle which is fluorescently labeled. Our
analytic results are in excellent agreement with the results of stochastic
simulations, which are performed using the Gillespie algorithm.
