Department of Mathematics

A long-standing open problem in statistical mechanics is the derivation of the Boltzmann equation from a Hamiltonian system. As a special example of such a system, consider the {\it hard sphere model.} in which there are $N$ spheres of diameter $\epsilon$ that travel according to their velocities and collide elastically. In a Boltzmann-Grad limit, we send $N\to\infty$, $\epsilon\to 0$ in such a way that $N\epsilon^{d -1}\to Z$ where $Z$ is a positive finite number.

If $f(x,v,t)$ denotes the density of the particles of velocity $v$, then $f$ satisfies the Boltzmann equation. I will discuss several variants of the hard sphere model and state some conjectures.