PDMP characterisation of event-chain Monte Carlo algorithms for particle systems

TL;DR

This paper characterizes event-chain Monte Carlo algorithms as piecewise deterministic Markov processes, proving their invariance and ergodicity, which advances the theoretical understanding and potential generalization of these algorithms for particle systems.

Contribution

It introduces a novel framework to analyze ECMC algorithms as PDMPs, enabling proofs of invariance and ergodicity and allowing for more flexible scheme designs.

Findings

Proved invariance of ECMC stationary distribution.

Established ergodicity for soft- and hard-sphere systems.

Enabled generalization of ECMC schemes.

Abstract

Monte Carlo simulations of systems of particles such as hard spheres or soft spheres with singular kernels can display around a phase transition prohibitively long convergence times when using traditional Hasting-Metropolis reversible schemes. Efficient algorithms known as event-chain Monte Carlo were then developed to reach necessary accelerations. They are based on non-reversible continuous-time Markov processes. Proving invariance and ergodicity for such schemes cannot be done as for discrete-time schemes and a theoretical framework to do so was lacking, impeding the generalisation of ECMC algorithms to more sophisticated systems or processes. In this work, we characterize the Markov processes generated in ECMC as piecewise deterministic Markov processes. It first allows us to propose more general schemes, for instance regarding the direction refreshment. We then prove the invariance of the correct stationary distribution. Finally, we show the ergodicity of the processes in soft- and hard-sphere systems, with a density condition for the latter.