cours-5

ENTROPY PRODUCTIONAND CONVERGENCE TO EQUILIBRIUMC. VILLANI, ENS LYONcvillani@umpa.ens-lyon.frThis set of notes was used to complement my short course on the convergenceto equilibrium for the Boltzmann equation, given at Institut Henri Poincar´e inNovember-December 2001, as part of the Hydrodynamic limits program organizedby Stefano Olla and Franc¸ois Golse. The informal style is in accordance withthe fact that this is neither a reference book, nor a research paper. The readercan use my review paper, A review of mathematical topics in collisional kinetictheory, as a reference source to dissipate any ambiguity with respect to notationfor instance. Apart from minor corrections here and there, the main changes withrespect to the original version of the notes were the addition of a final section topresentsomemorerecent developments andopendirections, andthechange of thesign convention for the entropy, to agree with physical tradition. Irene Mazzella iswarmly thanked for kindly typesetting a preliminary version of this manuscript.CONTENTS1. The entropy production problem for the Boltzmann equation 22. Tentative panorama 153. Reminders from information theory 184. Quantitative H theorem 255. Spatially homogeneous Boltzmann equation: state of the art 436. Related topics 497. Wiping out of spatial inhomogeneities 668. Towards exponential convergence? 7911. The entropy production problem for the Boltzmann equation.I shall start with Boltzmann’s brilliant ...