Freezing Transition in Very Small Systems of Hard Spheres

By applying a rigorous formalism, the grand distribution function of very small systems of hard spheres is obtained. The hard sphere freezing transition appears as two peaks of this function. The formalism requires input of the available volume, i.e., the configurationally averaged volume of a system that is available for an additional sphere center. These volumes are computed numerically. We show that by this treatment the freezing transition (1) follows “naturally,” i.e., properties of the fluid and the solid phases need not be inserted into the treatment in advance; (2) is already apparent in systems containing a number of spheres as small as eight; and (3) is caused by the system avoiding configurations that can best be characterized as “defective solids.”