Geometrical Considerations on Hard Core Problems

Abstract

A new formulation called stochastic geometry is proposed for hard core problems, in which short range correlations geometrically brought on because of the volume exclusion effect of the hard cores play a major role, and the system is described in terms of properly defined contiguous pair distribution functions (CPDF). A homogeneous integral equation for the CPDF is derived as the condition of stationary Markoff process. Application of this formulation to the hard disk system leads to a solution in a form of high density expansion without the assumption of the crystalline long range order. The equation of states, entropy and contiguous distribution functions are obtained. The pressure thus obtained is somewhat higher than the generally accepted values for the solid phase, and the entropy is larger. The possibility of disappearance of hard disk transition is discussed. Furthermore the possible differences of the melting transition due to the dimensionality are discussed through geometrical consideration.