## Abstract

The evolution of a gas can be described by different models
depending on the scale of observation. A natural question, raised by Hilbert
by Hilbert in his sixth problem, is to know if these models
provide consistent predictions. In the case of dilute gases of
hard spheres, Lanford showed in 1974 that the Boltzmann equation
appears as a law of large numbers in the limit where the number of particles
particles tends to infinity, at least for very short times. In
this talk, we will present Lanford's result and some more recent
extensions to understand the fluctuations and large deviations around the Boltzmann equation.