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.

Statistical Mechanics (cond-mat.stat-mech); Artificial Intelligence (cs.AI)

arXiv:2609.07620 [cond-mat.stat-mech]

(or arXiv:2609.07620v1 [cond-mat.stat-mech] for this version)

https://doi.org/10.48550/arXiv.2609.07620 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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