Understanding diffusion from first principles of physics has been a major challenge since the groundbreaking works of Einstein, Smoluchowski, Perrin and Langevin  in the early years of the XXth century. A simple but mathematically (and phenomenologically, as well) sufficiently rich model is the random Lorentz gas (proposed by H. Lorentz in 1905) where infinite-mass spherical scatterers are placed randomly (according to a Poisson Point Process) in R^d and point-like particles move in space observing Newton's laws: flying free between successive specular collisions on the infinite-mass put scatterers. Randomness comes with the initial conditions (the positions of the put scatterers and the initial velocities of the gas-particles) otherwise the dynamics is fully deterministic, Newtonian. One expects that under suitable scaling the trajectory of a gas particle looks like totally random motion (so-called "Brownian motion" of the mathematicians). However, since the dynamics is deterministic (only the initial conditions are random) this motion is by no means a usual random walk - for which these type of limit theorems are well understood. 

 

Decades ago great progress was achieved  [G. Gallavotti (1969-71), H. Spohn (1978), C Boldrighini, L. Bunimovich, Ya Sinai (1983)] in understanding the limiting behaviour under a particular low-density limit and finite time horizon. In recently published  work we were able to push these results well beyond, to time scales where the long memory of the system (due to physical and geometric causes) is manifestly present and causes complex difficulties. 

 

After a survey of the historical background I will outline the mathematical results. The main ingredients are a probabilistic coupling of the mechanical trajectory with a Markovian random fight process, and probabilistic and geometric controls on the efficiency of this coupling.

 

Based on joint work with Christopher Lutsko. Commun. Math. Phys. (2020).