Entropy and entropy production have recently become mathematical tools for kinetic and hydrodynamic limits, when deriving the macroscopic behaviour of systems from the interaction dynamics of their many microscopic elementary constituents at the atomic or molecular level.

During a special semester on Hydrodynamic Limits at the Centre Émile Borel in Paris, 2001 two of the research courses were held by C. Villani and F. Rezakhanlou. Both illustrate the major role of entropy and entropy production in a mutual and complementary manner and have been written up and updated for joint publication. Villani describes the mathematical theory of convergence to equilibrium for the Boltzmann equation and its relation to various problems and fields, including information theory, logarithmic Sobolev inequalities and fluid mechanics. Rezakhanlou discusses four conjectures for the kinetic behaviour of the hard sphere models and formulates four stochastic variations of this model, also reviewing known results for these.

From the reviews: "This nice book is based on two courses given, respectively, by Fraydoun Rezakhanlou and Cédric Villani at the Centre Émile Borel of the Institut Henri Poincaré in a special semester organized in the fall term of 2001 by Francois Golse and Stefano Olla. ? discusses many extensions, open questions and perspectives which should be valuable for many researchers in the field of the Boltzmann equation. ? This book ? will be useful to all mathematicians interested in entropy methods." (Clément Mouhot, Mathematical Reviews, Issue 2010 h)