Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem
Abstract
The recently proposed Microcanonical Hamiltonian Monte Carlo algorithm has not yet been studied in detail from a thermodynamic point of view; this work aims to fill that gap. We demonstrate how thermodynamical state variables and potentials can be derived and thereby demonstrate that the construction of the algorithm formally represents a microcanonical thermodynamic ensemble. In particular, we demonstrate (analytically and numerically) that the algorithm fulfils the Helmholtz theorem, an alternative formulation of the first law of thermodynamics. Furthermore, we construct a new sampling algorithm that extends the original to lower-dimensional inference problems. Finally, we argue that canonical Markov Chain Monte Carlo algorithms are more natural than Microcanonical Hamiltonian Monte Carlo from the thermodynamic and information-theoretic point of view.
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BibTeX
@misc{campe2026microcanonical,
title = {{Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem}},
author = {Heinrich von Campe and Bjoern Malte Schaefer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07620},
eprint = {2609.07620},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07620},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem
AU - Heinrich von Campe
AU - Bjoern Malte Schaefer
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07620
UR - https://arxiv.org/abs/2609.07620
AB - The recently proposed Microcanonical Hamiltonian Monte Carlo algorithm has not yet been studied in detail from a thermodynamic point of view; this work aims to fill that gap. We demonstrate how thermodynamical state variables and potentials can be derived and thereby demonstrate that the construction of the algorithm formally represents a microcanonical thermodynamic ensemble. In particular, we demonstrate (analytically and numerically) that the algorithm fulfils the Helmholtz theorem, an alternative formulation of the first law of thermodynamics. Furthermore, we construct a new sampling algorithm that extends the original to lower-dimensional inference problems. Finally, we argue that canonical Markov Chain Monte Carlo algorithms are more natural than Microcanonical Hamiltonian Monte Carlo from the thermodynamic and information-theoretic point of view.
ER -