Liquid-solid phase transition of a system with two particles in a rectangular box

We study the statistical properties of two hard spheres in a two-dimensional rectangular box. In this system, a relation similar to the van der Waals equation is obtained between the width of the box and the pressure working on the sidewalls. The autocorrelation function of each particle's position is calculated numerically. This calculation shows that, near the critical width, the time at which the correlation becomes zero gets longer as the height of the box increases. Moreover, fast and slow relaxation processes such as the alpha and beta relaxations in supercooled liquids are observed when the height of the box is sufficiently large. These relaxation processes are discussed with reference to the probability distribution of the relative positions of the two particles.