February 2024 Sampling Algorithms in Statistical Physics: A Guide for Statistics and Machine Learning
Michael F. Faulkner, Samuel Livingstone
Author Affiliations +
Statist. Sci. 39(1): 137-164 (February 2024). DOI: 10.1214/23-STS893


We discuss several algorithms for sampling from unnormalized probability distributions in statistical physics, but using the language of statistics and machine learning. We provide a self-contained introduction to some key ideas and concepts of the field, before discussing three well-known problems: phase transitions in the Ising model, the melting transition on a two-dimensional plane and simulation of an all-atom model for liquid water. We review the classical Metropolis, Glauber and molecular dynamics sampling algorithms before discussing several more recent approaches, including cluster algorithms, novel variations of hybrid Monte Carlo and Langevin dynamics and piece-wise deterministic processes such as event chain Monte Carlo. We highlight cross-over with statistics and machine learning throughout and present some results on event chain Monte Carlo and sampling from the Ising model using tools from the statistics literature. We provide a simulation study on the Ising and XY models, with reproducible code freely available online, and following this we discuss several open areas for interaction between the disciplines that have not yet been explored and suggest avenues for doing so.

Funding Statement

MFF acknowledges support from EPSRC fellowship EP/P033830/1. SL acknowledges support from EPSRC grant EP/V055380/1.


This work was conceived at the Scalable inference; statistical, algorithmic, computational aspects workshop (part of i-like, an EPSRC programme grant) at the Isaac Newton Institute for Mathematical Sciences, University of Cambridge. All simulations were performed on BlueCrystal 4 at the Advanced Computing Research Centre, University of Bristol.


Download Citation

Michael F. Faulkner. Samuel Livingstone. "Sampling Algorithms in Statistical Physics: A Guide for Statistics and Machine Learning." Statist. Sci. 39 (1) 137 - 164, February 2024. https://doi.org/10.1214/23-STS893


Published: February 2024
First available in Project Euclid: 18 February 2024

MathSciNet: MR4718531
Digital Object Identifier: 10.1214/23-STS893

Keywords: event chain Monte Carlo , Glauber dynamics , hard-disk model , hybrid Monte Carlo , Ising model , Langevin dynamics , Markov chain Monte Carlo , Metropolis , molecular dynamics , molecular simulation , Potts model , sampling algorithms , statistical physics , XY model

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.39 • No. 1 • February 2024
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