Thermodynamically small systems, with a number N of interacting particles in the range of 1–1000, are increasingly of interest in science and engineering. While the thermodynamic formalism for bulk systems, where N approaches infinity, was established long ago, the thermodynamics of small systems is currently approached by adding new variables in a somewhat ad hoc fashion. We propose a more rigorous approach based on machine learning (ML), which we demonstrate by applying both unsupervised (diffusion maps, autoencoders) and supervised (classical neural networks) ML methods to large data sets from Monte Carlo simulations of systems comprising $N=3$ Lennard-Jones particles at fixed temperature. The ML methods clearly identify structural and energetic changes that occur in this model system and suggest that the data may be collapsed from the original nine dimensions to two. Using intuition and screening, we identified two simple geometric properties of the system as a useful variable set.

热力学小的系统，相互作用的粒子数N在1–1000范围内，在科学和工程学中越来越受到关注。虽然N接近无穷大的散装系统的热力学形式主义是很早以前就建立的，但目前通过以某种临时的方式添加新变量来接近小型系统的热力学。我们提出了一种基于机器学习（ML）的更为严格的方法，通过将无监督（扩散图，自动编码器）和有监督（经典神经网络）机器学习方法应用于来自以下系统的蒙特卡洛模拟的大型数据集，我们对此进行了演示$ñ=3$Lennard-Jones颗粒在固定温度下。ML方法清楚地识别了此模型系统中发生的结构性变化和能量变化，并建议将数据从原始的9个维度折叠为2个维度。使用直觉和筛选，我们将系统的两个简单几何特性识别为有用的变量集。