Connectedness of the configuration space of hard disk systems

Hard disk systems are often regarded as prototypes for simple fluids, and can offer valuable insight into the origins of phase transitions and glass transitions in general. A phase transition is usually believed to occur when averages of thermodynamic quantities over the accessible part of the configuration space change discontinuously. A phase transition could then be driven by a discontinuous change in the accessible region, and this depends on the connectedness of the configuration space as a function of disk radius. A regularized potential energy function is defined on the configuration space, the critical points of which correspond to configurations where the topology of the configuration space changes, and are extensively sampled for small numbers of disks. Knowledge of the critical points allows the configuration space to be decomposed into regions that are homeomorphic to spheres of known dimensions, but does not indicate how these regions are attached to one other. Preliminary attaching maps are then found using the zero-temperature string method, resulting in a complete characterization of the topology of the configuration space of hard disk systems for small numbers of disks.