A simple model of gas is the hard spheres model. It is a billiard of little particles which can interact very strongly at very small distance (think for example of real billiards with a lot of balls). Because understanding such system is an outstanding problem, people tried to find a limiting process. A first equation governing the density of one particle was given by Boltzmann :
\[\partial_t f +v\cdot\nabla_x f = Q(f,f).\]
In its formal derivation Boltzmann supposed that two different particles are almost independent, so the probability of having two particles at the same place is the the product of probability. The validity of such equation is a priori not clear since it adds some irreversibly that does not exist in the hard sphere model.
Lanford solved the problem in its ‘75 paper: Boltzmann’s equation is true, up to a time independent of the number of particles (however each particle will have in mean less than one collision).
Now comes the question of the boundary. We expect to find some “Lanfords” theorem even if we add some boundary condition. A first example are the specular reflections, for a deterministic law. An other example, which would be very important in physics, is the evolution of a gas between two hot plaques. Then the reflection condition is stochastic. I am interested in a third type of reflection, also stochastic, which is a modeling of a rough boundary.
During my talk I will present some ideas of the proof of Boltzmann in the full space and the adaptation in the case with boundary.