# Engel et al 2013

### From Werner KRAUTH

M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, E. P. Bernard, W. Krauth *Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods* Phys. Rev. E 87, 042134 (2013)

## Contents |

# Paper

**Abstract**: 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. Phase coexistence is visualized for individual
configurations via the local orientations, and positional correlation functions
are computed. Our results confirm the first-order nature of the liquid-hexatic
phase transition in hard disks.

Electronic version (from arXiv)

Paper in Physical Review E (Subscription needed)

# Context

This paper, on a **first**-order transition in **two** dimensions, by a collaboration on **three** continents (!),
confirms the findings of my 2011 paper, with Etienne Bernard about the first-order
liquid-hexatic transition in hard disks. Working on the paper was extremely interesting and our earlier
calculations were put to a quite stringent test. Not only were the simulation algorithms different (massively parallel Monte Carlo, event-driven molecular dynamics, event-driven Monte Carlo), but also the approaches to determine the equation of state varied considerably. In Monte Carlo, the pressure is obtained by an extrapolation of the pair-correlation function, which needs a lot of fine-tuning. In Molecular dynamics, one simply computes the number of collisions for as long a simulation as possible. Fortunately, all came out, finally, as we had said in 2011.