Hierarchical sorting is a fundamental task for programmable matter, inspired by the spontaneous formation of interfaces and membranes in nature. The task entails particles of different types, present in fixed densities, sorting into corresponding regions of a space that are themselves organized. By analyzing the Gibbs distribution of a general fixed-magnetization model of equilibrium statistical mechanics, we prove that particles moving stochastically according to local affinities solve the hierarchical sorting task. The analysis of fixed-magnetization models is notoriously difficult, and approaches that have led to recent breakthroughs in sampling the low-temperature regime only work in the variable-magnetization setting by default. To overcome this barrier, we introduce a new approach for comparing the partition functions of fixed- and variable-magnetization models. The core technique identifies a special class of configurations that contribute comparably to the two partition functions, which then serves as a bridge between the fixed- and variable-magnetization settings. Our main result is an estimate of the Gibbs distribution that unifies existing and new results for models at fixed magnetization, including the Ising, Potts, and Blume--Capel models, and leads to stochastic distributed algorithms for hierarchical sorting and other self-organizing tasks, like compression and separation.

翻译：分层排序是受自然界中界面和膜自发形成启发的可编程物质的一项基本任务。该任务要求不同类型的粒子以固定密度存在，并排序到空间中相应且自身有序的区域。通过分析平衡统计力学中一般固定磁化模型的吉布斯分布，我们证明了粒子根据局部亲和性随机运动能够解决分层排序任务。固定磁化模型的分析历来困难，近期在低温区域采样方面取得突破的方法默认仅适用于可变磁化场景。为克服这一障碍，我们提出了一种比较固定磁化与可变磁化模型配分函数的新方法。核心技术在于识别一类对两种配分函数贡献相当的特定构型，从而在固定磁化与可变磁化设定之间建立桥梁。我们的主要结果是对吉布斯分布的估计，该结果统一了现有及新的固定磁化模型（包括Ising、Potts和Blume-Capel模型）的相关结论，并推导出适用于分层排序及其他自组织任务（如压缩与分离）的随机分布式算法。