Study of the positional and orientational contributions to the Helmholtz free energy of a finite hard-disk system. A molecular dynamics analysis of its hexatic transition

Based on an equilibrium thermodynamic scheme, this work studies the separation of the Helmholtz free energy of a hard-disk system into its positional and orientational contributions. For the positional part, an accurate two dimensional version of the Carnahan-Starling equation of state is proposed, while a simple ansatz of the one particle partition function for the orientational part is proposed. The latter expression, which contains information about the probability of the norm of the six-fold orientational order parameter, is obtained through molecular dynamics simulations. Perhaps due to finite size effects, the ordering of the fluid is not a sudden phenomenon: the fluid-hexatic phase transition is preceded by a linear increase in the orientational contribution to the pressure of the fluid. A similar result is obtained for the hexatic-solid phase transition. Thus, the hexatic first-order phase transition is located between two linear ordering processes, consistent with the KTHNY theory. The proposed separation also predicts the existence of first order fluid-hexatic phase transition by using two procedures: the first, based on a purely bimodal probability density function, produces a pressure with a van del Waals-like first-order phase transition, while the second, considering density fluctuations, results in a flattened pressure phase transition.