Diffusion of two particles with a finite interaction potential in one dimension

We investigate the dynamics of two interacting diffusing particles in an infinite effectively one-dimensional system; the particles interact through a steplike 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 have 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.