Abstract

Since the pioneering works of Hendrik Lorentz (1905) and Paul and Tatiana Ehrenfest (1912) the deterministic (Hamiltonian) motion of a point-like particle exposed to the action of a collection of fixed, randomly located short range scatterers has been a much studied model of physical diffusion under fully deterministic (Hamiltonian) dynamics, with random initial conditions. This model of physical diffusion is known under the name of “random Lorentz gas” or “random wind-tree model”. Celebrated milestones on the route to better mathematical understanding of this model of true physical diffusion are the Kinetic Limits for the tagged particle trajectory under the so-called Boltzmann-Grad (a.k.a. low density), or weak coupling approximations [Gallavotti (1970), Spohn (1978), Boldrighini-Bunimovich-Sinai (1982), respectively, Kesten-Papanicolaou (1980)]. Under a second diffusive space-time scaling limit – done as a second step, after the kinetic approximations- the central limit theorem (CLT) and invariance principle (IP) for the tagged particle motion follow. However, the CLT /IP under bare diffusive space-time scaling (without first applying the kinetic approximations) remains a Holy Grail. In recent work we have obtained some intermediate results, partially interpolating between the two-steps-limit (first kinetic, then diffusive – as described above) and the bare-diffusive-limit (Holy Grail). We establish the Invariance Principle for the tagged particle trajectories under a joint kinetic+diffusive limiting procedure, performed simultaneously rather than successively, reaching significantly longer time scales than in earlier works. The Holy Grail (i.e., CLT under bare diffusive scaling) remains, however, beyond reach.  I will prersent a survey of the main problems and (historic and more recent) results, accessible for a broad range of mathematicians.