One
of the fundamental and initially controversial
theories of classical physics is Boltzmann's
kinetic theory of gases. Instead of tracking
the individual motions of billions of atoms,
it describes the evolution of the probability
that a particle has a certain position and velocity.
The equilibrium probability distributions have
been known for more than a hundred years, but
it has been very difficult to understand whether
and how fast convergence to equilibrium occurs.

Villani
(in collaboration with Desvillettes) obtained
the first results on the convergence rate for
initial data not close to equilibrium. In joint
work with Mouhot, he established nonlinear Landau
damping for the kinetic equations of plasma
physics, settling a long-standing debate. He
is one of the pioneers in applications of optimal
transport theory to geometric and functional
inequalities.