RESUMO
We utilized a blend of replica exchange and event-chain Monte Carlo techniques to generate candidate configurations, aiming for a maximal packing fraction of hard disks within a circular enclosure. Our investigation encompassed systems comprising N particles, with N ranging from 300 to 720. Through our analysis, we identified 108 novel maximal packings, with some surpassing existing configurations by over 0.001 in packing fraction. As such, Monte Carlo methods demonstrate their efficacy in tackling optimization challenges of this nature.
RESUMO
We have explored the idea of producing the equilibrium equation of state, i.e. the pressure as a function of packing fraction, ßP(φ), of a confined system up to very high pressures to yield the configuration that leads to the maximum packing fraction. For this purpose we have massively implemented the replica exchange Monte Carlo algorithm in graphics processing units (GPUs), in such a way that each GPU core handles a single simulation cell. This yields a very easy scheme to implement parallelization for a very large amount of replicas (thousands), which densely sample configuration space. We have tested this idea with a very well studied system, i.e. discs confined in a circular cavity, for a number of particles N ≤ 125. In all cases, our outcomes for configurations having maximum packing fractions are in perfect agreement with those already reported and conjectured optimal in the literature, for which there is no formal mathematical proof, strongly suggesting that they are indeed optimal configurations. Furthermore, in most cases, we have obtained the same function ßP(φ), by compressing loose random configurations and by decompressing copies of the configuration having the largest packing fraction. This reveals numerically that the so obtained maximum packing configurations are the correct answer.