Prediction of a Structural Transition in the Hard Disk Fluid

TL;DR

This paper develops a new analytical method to study correlation decay in hard disk fluids, predicting a structural transition at high density consistent with previous simulations and identifying conditions where oscillations dampen or vanish.

Contribution

The paper introduces a novel approach based on the BBGKY hierarchy to analyze correlation decay and predict structural transitions in hard particle fluids.

Findings

Exponential damping of correlations occurs up to a packing fraction of ~0.718.

Predicted transition point aligns with Monte Carlo simulation estimates.

No structural transition is found in hard rods at any density below close packing.

Abstract

Starting from the second equilibrium equation in the BBGKY hierarchy under the Kirkwood superposition closure, we implement a new method for studying the asymptotic decay of correlations in the hard disk fluid in the high density regime. From our analysis and complementary numerical studies, we find that exponentially damped oscillations can occur only up to a packing fraction {\eta}*~0.718, a value which is in substantial agreement with the packing fraction, {\eta}~0.723, believed to characterize the transition from the ordered solid phase to a dense fluid phase, as inferred from Mak's Monte Carlo simulations [Phys. Rev. E 73, 065104 (2006)]. We next show that the same method of analysis predicts that exponential damping of oscillations in the hard sphere fluid becomes impossible when \lambda = 4n\pi {\sigma}^3 [1 + H(1)]>/- 34.81, where H(1) is the contact value of the correlation function, n is the number density and {\sigma} is the sphere diameter, in exact agreement with the condition, \lambda >/- 34.8, first reported in a numerical study of the Kirkwood equation by Kirkwood et al. [J. Chem. Phys. 18, 1040 (1950)]. Finally, we show that our method confirms the absence of any structural transition in hard rods for the entire range of densities below close packing.