Free energy landscapes for a small number of hard disks

We investigate free energy landscapes in a simple system consisting of a small number (N < 9) of disks moving in a circular corral. For a given number of disks, preferred configurations of the disks exist where the system spends most of its time. There are occasional rearrangements by which the system switches from one configuration to a topologically distinct configuration. We find that these rearrangements can be simplified: in each case, a topological change involves one particle moving between two others. This observation allows us to reduce the problem to a one-dimensional free energy landscape. The rare states where one particle is exactly between two others corresponds to a high entropic barrier, and the barrier height relates to the rearrangement time scale as expected. Decreasing the system size causes the rearrangements to take dramatically longer time scales, analogous to a glass transition. Our studies of N = 3 to 8 give us insight into the nature of free energy landscapes for N >> 10 particles.