# Machine Learning the H-theorem

arXiv: https://arxiv.org/abs/2508.14003 · HTML: https://mlsanity.com/paper/2508.14003/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 50 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Builds on (most cited first, 2 of 2)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Learning the Arrow of Time](https://mlsanity.com/paper/1907.01285/) (2019)

## Abstract (arXiv)

The H-theorem provides a microscopic foundation for the Second Law of Thermodynamics and therefore occupies a central place in statistical physics. At the same time, its relation to microscopic reversibility has remained conceptually subtle. To investigate how an arrow of time may be inferred directly from microscopic data, we study the relaxation of randomly initialized hard disks in a periodic box. We construct a permutation-invariant neural network based on the DeepSets architecture. The model is trained only to assign later states a larger scalar value than earlier states. We compare the learned scalar with the Boltzmann H-functional and assess to what extent the dynamics alone lead the model toward the structure implied by the H-theorem.