Hard-disk computer simulations -- a historic perspective

TL;DR

This paper reviews the historical and recent computational efforts to estimate the pressure in the hard-disk model, highlighting the challenges and presenting new results and algorithms for large systems.

Contribution

It introduces new unbiased pressure estimators, compares sampling algorithms, and provides benchmark data for large-scale hard-disk simulations.

Findings

Pressure estimates up to one million disks

New unbiased pressure estimators introduced

Sampling algorithms and pressure data made openly available

Abstract

We discuss historic pressure computations for the hard-disk model performed since 1953, and compare them to results that we obtain with a powerful event-chain Monte Carlo and a massively parallel Metropolis algorithm. Like other simple models in the sciences, such as the Drosophila model of biology, the hard-disk model has needed monumental effort to be understood. In particular, we argue that the difficulty of estimating the pressure has not been fully realized in the decades-long controversy over the hard-disk phase-transition scenario. We present the physics of the hard-disk model, the definition of the pressure and its unbiased estimators, several of which are new. We further treat different sampling algorithms and crucial criteria for bounding mixing times in the absence of analytical predictions. Our definite results for the pressure, for up to one million disks, may serve as benchmarks for future sampling algorithms. A synopsis of hard-disk pressure data as well as different versions of the sampling algorithms and pressure estimators are made available in an open-source repository.