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The Work of Cédric

The starting point of Cédric Villani’s work goes back to the introduction of entropy in the nineteenth century by L. Carnot and R. Clausius. At the time, entropy was a vague concept and its rigorous definition had to wait until the fundamental work of L. Boltzmann who introduced nonequilibrium statistical physics and the famous H functional. Boltzmann’s work, though a fundamental breakthrough, did not resolve the question concerning the nature of entropy and time arrow; the debate on this central question continued for a century until today. J. von Neumann, in recommending C. Shannon to use entropy for his uncertainty function, quipped that entropy is a good name because “nobody knows what entropy really is, so in a debate you will always have the advantage”. The first result of Villani I will report on concerns the fundamental connection between entropy and its dissipation. In this work, we will see that rigorous mathematical analysis is not just a display of powerful analytic skill, but also leads to deep insights into nature. Based on this work, Villani has developed a general theory, hypercoercivity, which applies to broad systems of equations. In a separate direction, entropy was used by Villani as a fundamental tool in optimal transport and the study of curvature in metric spaces. Finally, I will describe Villani’s work on Landau damping, which predicts a very surprising decay (and thus the word damping) of the electric field in a plasma without particle collisions, and therefore without entropy increase. This is in sharp contrast with Boltzmann’s picture that the time irreversibility comes from collision processes.