Sampling from a polytope and hard-disk Monte Carlo

Abstract

The hard-disk problem, the statics and the dynamics of equal two-dimensional hard spheres in a periodic box, has had a profound influence on statistical and computational physics. Markov-chain Monte Carlo and molecular dynamics were first discussed for this model. Here we reformulate hard-disk Monte Carlo algorithms in terms of another classic problem, namely the sampling from a polytope. Local Markov-chain Monte Carlo, as proposed by Metropolis et al. in 1953, appears as a sequence of random walks in high-dimensional polytopes, while the moves of the more powerful event-chain algorithm correspond to molecular dynamics evolution. We determine the convergence properties of Monte Carlo methods in a special invariant polytope associated with hard-disk configurations, and the implications for convergence of hard-disk sampling. Finally, we discuss parallelization strategies for event-chain Monte Carlo and present results for a multicore implementation.

References in corpus (3)

• Shortcomings of the Bond Orientational Order Parameters for the Analysis of Disordered Particulate Matter
• Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods
• Massively parallel Monte Carlo for many-particle simulations on GPUs

Cited by in corpus (14)

• 2D Melting: From Liquid-Hexatic Coexistence to Continuous Transitions
• Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps
• Hard-sphere melting and crystallization with event-chain Monte Carlo
• Event-chain Monte Carlo: foundations, applications, and prospects
• Event-chain Monte Carlo for classical continuous spin models
• Fast MCMC sampling algorithms on polytopes
• All-atom computations with irreversible Markov chains
• Efficient Irreversible Monte Carlo samplers
• JeLLyFysh-Version1.0 -- a Python application for all-atom event-chain Monte Carlo
• Parallelized event chain algorithm for dense hard sphere and polymer systems
• Multithreaded event-chain Monte Carlo with local times
• Large-scale dynamics of event-chain Monte Carlo
• Sparse hard-disk packings and local Markov chains
• Computational Methods toward Ultrastable Glasses