From particle systems to the Boltzmann equation
In his 6th problem, Hilbert asks whether it is possible to justify rigourously the mechanisms leading from classical, molecular dynamics to fluid motion. The idea is to use the Boltzmann equation as an intermediate description. In this talk we shall discuss the derivation of the Boltzmann equation, which can be seen as the limit of a law of large numbers for the many-particle system, and study fluctuations around this limit. This will in particular enable us to understand the appearance of irreversibility in the process; we shall also see how some (linear) fluid equations can be derived in this manner.This corresponds to joint works with Thierry Bodineau, Laure Saint-Raymond and Sergio Simonella.