  A probabilistic Markov Chain (MC) surrogate model for a two-dimensional
system of interacting particles within a square domain having inherent
symmetries is developed. Particles are assumed to be circular and identical,
colliding with each other and with rigid domain walls via perfectly elastic
collisions. Simulation results over many realizations are used to develop and
evaluate the surrogate MC model. The stationary quantity of interest (QoI) is
the number of particles within each of nine coarser subdomains, delineated via
three internal geometric symmetries. The surrogate model is used to quantify
QoI mean properties and uncertainty as the particle radius is varied.
