  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.
