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.

A simple and efficient algorithm of the molecular-dynamics simulation of the hard disk system based on the Event-Driven method is developed. From the analysis of algorithm, the complexity is O(log N ) per 1 event, and the constant coefficient of the complexity is smaller than conventional efficient algorithm based on the concept of Cell-Crossing Event. The maximum performance of more than 460 millions of collisions per CPU-hour on the Alpha600 compatible in a 2500 particle system is achieved. An extension to the infinite-space system based on this algorithm is also proposed.

Alder and Wainwright discovered the slow power decay ~t(-d/2) (d is dimension) of the velocity autocorrelation function in moderately dense hard-sphere fluids using the event-driven molecular dynamics simulations. In the two-dimensional (2D) case, the diffusion coefficient derived using the time correlation expression in linear response theory shows logarithmic divergence, which is called the "2D long-time-tail problem." We reexamined this problem to perform a large-scale, long-time simulation with 1x10(6) hard disks using a modern efficient algorithm and found that the decay of the long tail in moderately dense fluids is slightly faster than the power decay (~1/t) . We also compared our numerical data with the prediction of the self-consistent mode-coupling theory in the long-time limit [~1/(t sqrt[ln t])] .

We simulate crystallization and melting with local Monte Carlo (LMC), event-chain Monte Carlo (ECMC), and with event-driven molecular dynamics (EDMD) in systems with up to one million three-dimensional hard spheres. We illustrate that our implementations of the three algorithms rigorously coincide in their equilibrium properties. We then study nucleation in the NVE ensemble from the fcc crystal into the homogeneous liquid phase and from the liquid into the homogeneous crystal. ECMC and EDMD both approach equilibrium orders of magnitude faster than LMC. ECMC is also notably faster than EDMD, especially for the equilibration into a crystal from a disordered initial condition at high density. ECMC can be trivially implemented for hard-sphere and for soft-sphere potentials, and we suggest possible applications of this algorithm for studying jamming and the physics of glasses, as well as disordered systems.