Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem

• URL: http://arxiv.org/abs/2609.07620v1
• Date: Mon, 07 Sep 2026 15:24:11 GMT
• Title: Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem
• Abstract summary: We show how thermodynamical state variables and potentials can be derived and thereby demonstrate that the construction of the algorithm formally represents a microcanonical thermodynamic ensemble.<n>We construct a new sampling algorithm that extends the original to lower-dimensional inference problems.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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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