Brownian Motion in Hamiltonian systems
Brownian motion happens everywhere in nature. The conceptually simplest
example is of course the pollen grains observed by R. Brown, in
1827. Since the work of Einstein and Smoluchowksi (1905-1906), we
understand that it is the chaotic motion of solvent molecules that
drives the Brownian motion of microparticles.
Nevertheless, a rigorous proof that Brownian motion actually takes place in a system that is described microscopically by deterministic equations of motion (like Brown's microparticles!) has turned out to be a very difficult problem.
In collaboration with L. Erdös and H-T. Yau, we have proven the onset of diffusive behaviour for random Schrödinger operators, i.e. diffusive behaviour on a time scale that depends on the (small) interaction strength. The short review [ESY] also contains the references to our original papers.
The major goal is to prove diffusion up to infinite times.
Here our contribution was to find a model where one can give a
proof without performing any scaling limits. This
model consists of a heavy quantum particle hopping on a lattice and
interacting with a phonon field. See our preprint
Recently, the result has been strenghtened (extended to dimension 3, with very little infrared regularity required) in collaboration with A. Kupiainen (work in progress).