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Obtaining a rigorous justification of the kinetic theory of gases fromNewton's second law of dynamics for a large number of identical spheresinteracting by elastic binary collisions, is a problem formulated by Hilbert in1900 (Hilbert's 6th problem). In 1975, Lanford demonstrated the validity of theBoltzmann equation over a very short time interval, of the order of a fractionof the average time between two successive collisions experienced by the sameparticle. This result of Lanford can be interpreted as a kind of law of largenumbers as the number of particles tends to infinity. This point of view raisesseveral questions. First, the core of the argument used by Boltzmann to arriveat the equation bearing his name is the assumption that two particles about tocollide are statistically almost independent. This suggests to examine thevalidity of this hypothesis by studying the dynamics of correlations betweenparticles. On the other hand, the interpretation of the Boltzmann equation as alaw of large numbers leads to study precisely the fluctuations of the phasespace empirical measure of the particle system about its mean (whose evolutionis described by the Boltzmann equation). A series of recent papers by T.Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella answers these variousquestions and allows going beyond the Boltzmann equation in the understandingof the kinetic theory of gases.

Validit\'e de la th\'eorie cin\'etique des gaz: au-del\`a de  l'\'equation de Boltzmann [d'apr\`es T. Bodineau, I. Gallagher, L.  Saint-Raymond, S. Simonella]