arXiv AI September 10, 2026 By Heinrich von Campe, Bjoern Malte Schaefer

Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem

The paper investigates the Microcanonical Hamiltonian Monte Carlo (MHMC) algorithm from a thermodynamic perspective, deriving its state variables and potentials to show that it represents a microcanonical thermodynamic ensemble. It demonstrates analytically and numerically that MHMC satisfies the Helmholtz theorem, an alternative form of the first law of thermodynamics, and introduces a new sampling algorithm tailored for lower-dimensional inference problems. The authors conclude that canonical Markov Chain Monte Carlo methods are more natural than MHMC when evaluated through thermodynamic and information-theoretic lenses.

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