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Title : The Random Lorentz Gas - Invariance Principle Beyond the Kinetic Time Scales
Abstract :
Since the pioneering work of 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)]. Once the Kinetic Limits are established, under a second diffusive space-time scaling limit the central limit theorem (CLT) and invariance principle (IP) follows. 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 any earlier result. The main ingredient is a coupling of the Hamiltonian trajectory (with random initial conditions) and an approximate Markovized version of the motion, and probabilistic and geometric controls on the efficiency of this coupling. The Holy Grail remains, however, beyond reach. </p>
In the colloquium talk I will present a concise historic survey of the problems and some insight regarding the main ideas of the more recent work.
Plus de détails dur le site web The Colloquium ici : https://www.ceremade.dauphine.fr/~salez/colloquium.html
