van der Waals loops and the melting transition in two dimensions

Evidence for the existence of van der Waals loops in pressure p versus volume v plots has for some time supported the belief that melting in two dimensions (2D) is a first-order phase transition. We report rather accurate equilibrium p(v) curves for systems of hard disks obtained from long Monte Carlo simulations. These curves, obtained in the constant volume ensemble, using periodic boundary conditions, exhibit well-defined van der Waals loops. We illustrate their existence for finite systems that are known to undergo a continuous transition in the thermodynamic limit. To this end, we obtain magnetization m versus applied field curves from Monte Carlo simulations of the two-dimensional Ising model, in the constant m ensemble, at the critical point. Whether van der Waals loops for disk systems behave in the L-->infinity limit as they do for the two-dimensional Ising model at the critical point cannot be ruled out. Thus, the often made claim that melting in 2D is a first-order phase transition, based on the evidence that van der Waals loops exist, is not sound. [S1063-651X(99)01603-7].