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

TL;DR

This paper provides an exact analytical solution for the dynamics of two interacting diffusing particles in one dimension with a finite potential, including probabilities of overtaking and trapping, validated by stochastic simulations.

Contribution

It introduces an exact solution to the 2+1-dimensional Fokker-Planck equation for two particles with finite interaction potential, including new probability measures.

Findings

Exact two-particle PDF derived for arbitrary initial conditions.

Overtake probability and trapping probability computed analytically.

Analytical results validated by Gillespie algorithm simulations.

Abstract

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.