[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む27 [無断転載禁止]©2ch.net (517ﾚｽ)
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228(1): 現代数学の系譜11 ガロア理論を読む 2017/01/02(月)08:33 ID:MUXssChK(12/59) AAS
つづき

1 Introduction
The starting point of Cedric 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.

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