Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods

We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.

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Figure 1

(a) Pair-correlation function $g\left(r\right)$ close to contact for $N={512}^2$ at density $\eta =0.698$ using LMC. Error bars are computed through 64 independent simulations. $g\left(r\right)$ is fitted with a fourth-order polynomial. Difference between $g\left(r\right)$ and the polynomial fit with the single-precision data (b) and double-precision data (c) for the histogram.

Figure 2

Autocorrelation function of the global orientation order parameter ${\Psi }_6\left(t\right)$ for $N={512}^2$, $\eta =0.698$ obtained with LMC, EDMD, ECMC, and MPMC. (a) Time is measured in number of attempted displacements (or collisions) per disk. (b) Time is measured in CPU or GPU hours.

Figure 3

Equation of state from ECMC, EDMD, and MPMC for $N={256}^2$ and $N={512}^2$. Error bars are mostly smaller than the symbols. Results agree within one standard deviation. The inset shows the relative pressure difference $\Delta P/P$ of EDMD and MPMC with respect to ECMC for $N={256}^2$.

Figure 4

Equation of state from ECMC and MPMC for $N={1024}^2$. Error bars are mostly smaller than the symbols. Results agree within one standard deviation. The inset shows the relative pressure difference $\Delta P/P$ of MPMC with respect to ECMC.

Figure 5

Orientational order parameter field $\psi \left(\mathbf{x}\right)$ of configurations obtained with the MPMC algorithm for system size $N={1024}^2$. With increasing density, (a) pure liquid ($\eta =0.700$), (b) a bubble of hexatic phase ($\eta =0.704$), and (c) a stripe regime of hexatic phase ($\eta =0.708$) are visible. The interface between the liquid and the hexatic phase is extremely rough. (d) A scale bar illustrates the size of the fluctuations. The phase of $\psi$ is represented via the color wheel.

Figure 6

Positional order parameter field $\chi \left(\mathbf{x}\right)$ of configurations obtained with (a), (b) the MPMC and (c), (d) ECMC algorithms for system size $N={1024}^2$. (a), (c) In the hexatic phase ($\eta =0.718$), positional order is short-range. (b), (d) Toward higher density ($\eta =0.720$), fluctuations are much weaker and bounded, as expected for a continuous transition to a solid phase. The scale bar and the color code for the phase of $\chi$ are identical to the scale bar and the color code in Fig. 5.

Figure 7

The positional correlation function $C_{{\mathbf{q}}_0}\left(r\right)$ shows exponential decay at density $\eta =0.718$ and approaches a power law $\propto r^{-1/3}$ at $\eta =0.720$. (a) System size $N={512}^2$: Excellent agreement between ECMC and MPMC. (b) System size $N={1024}^2$: Excellent agreement in the hexatic phase ($\eta =0.718$) and fair agreement at the approach of the solid phase ($\eta =0.720$). Our algorithms fall out of strict equilibrium in the solid and long-scale correlations become sensitive to the boundary conditions.

Figure 8

Influence of the configuration averaging on the positional configuration functions in the hexatic phase. For single configurations ($n=1$, data incoherently averaged), positional correlations cannot decay below a level given by the square root of the ratio of the correlated domain size to the system size. Coherent averaging over $n=2$, 4, 8, 16, and 32 configurations reduces the noise level and the correlations. The data corresponds to a single long MPMC run with $N={1024}^2$ particles at density $\eta =0.718$.