From Boltzmann to random matrices and beyond

Djalil Chafaï (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 IUF - Institut universitaire de France

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.

Mots clés: Entropy; Fisher information; Boltzmann; Shannon; Voiculescu; Markov processes; Diffusion processes; Central limit theorem; Free probability; Random matrices; Ginibre ensemble; Circular law; Coulomb gas; Riesz kernel; Singular repulsion; Potential theory; Electrostatics; Equilibrium measure; Interacting Particle Systems; Mean-field interaction; Large Deviations Principle; Collective phenomena

Domaines: Histoire et perspectives sur les mathématiques [math.HO]; Probabilités [math.PR]; Physique mathématique [math-ph]; Analyse fonctionnelle [math.FA]

Djalil Chafaï. From Boltzmann to random matrices and beyond. 2014. ⟨hal-00987177v2⟩