From Boltzmann to random matrices and beyond

TL;DR

This paper explores the concept of entropy across various fields, connecting classical and modern theories including Boltzmann's work, free probability, random matrices, and mean-field systems, in an informal and interdisciplinary manner.

Contribution

It provides an expository overview linking entropy concepts from kinetic theory to random matrix theory and mean-field particle systems, fostering cross-disciplinary understanding.

Findings

Connections between entropy and free probability theory.

Analysis of the circular law in random matrices.

Insights into mean-field particle systems with external fields.

Abstract

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.