# Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem

arXiv: https://arxiv.org/abs/2609.07620 · HTML: https://mlsanity.com/paper/2609.07620/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 7 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Microcanonical Langevin Ensembles: Advancing the Sampling of Bayesian Neural Networks](https://mlsanity.com/paper/2502.06335/) (2025)
- [Hamiltonian Dynamics with Non-Newtonian Momentum for Rapid Sampling](https://mlsanity.com/paper/2111.02434/) (2021)
- [Fluctuation without dissipation: Microcanonical Langevin Monte Carlo](https://mlsanity.com/paper/2303.18221/) (2023)
- [Field-Level Inference with Microcanonical Langevin Monte Carlo](https://mlsanity.com/paper/2307.09504/) (2023)

## Abstract (arXiv)

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.