从玻尔兹曼到随机矩阵及其他
From Boltzmann to random matrices and beyond
摘要：
这些简要的注释建议跨领域地遵循熵概念的某些方面。从玻尔兹曼在气体动力学理论中的工作开始，访问了各种宇宙，包括马尔可夫过程及其亥姆霍兹自由能，中心极限定理中的香农单调性问题，Voiculescu自由概率论和自由中心极限定理，正则树上的随机游动，随机矩阵的复Ginibre系综的循环律，最后是任意维平均场粒子系统的渐近分析，受外场约束，并经历奇异对排斥。文本是写在一个非正式的风格驱动的能量和熵。它的目的是娱乐，并提供给好奇的读者在文学的切入点，并跨越边界的连接。
英文摘要：
These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, the Voiculescu free probability theory and the free central limit theorem, random walks on regular trees, the circular law for the complex Ginibre ensemble of random matrices, and finally the asymptotic analysis of mean-field particle systems in arbitrary dimension, confined by an external field and experiencing singular pair repulsion. The text is written in an informal style driven by energy and entropy. It aims to be recreative and to provide to the curious readers entry points in the literature, and connections across boundaries.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
