Machine Learning the H-theorem

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.