Title: Abnormal behavior in periodic Lorentz gases with infinite horizon
Speaker: Professor Nikolai Chernov
Speaker Info: University of Alabama at Birmingham
Brief Description:
Special Note:
Abstract:
(joint work of N. Chernov and D. Dolgopyat)Date: Tuesday, April 28, 2009Periodic Lorentz gas is a mathematical model of a gas of electrons in metals. A point(electron) moves freely and bounces off a periodic array of fixed round obstacles (molecules). Hyperbolicity, ergodicity, and basic statistical properties (exponential decay of correlations) have been established for this model. When the horizon is finite (i.e. the free path between collisions is bounded), then the particle exhibits a regular diffusion and, under a small external field, a regular current satisfying classical Ohm's law. We study periodic Lorentz gases with infinite horizon. This model exhibits abnormal diffusion. This fact was partially proved by Bleher in 1992 and completely proved in discrete time by Szasz and Varju in 2007; we prove it in real (continuous) time and derive further limit laws. We also show that under a small external field, the current is abnormal, too. We derive formulas for the current and relate it to the diffusion in the absence of the field. In physical terms, we show that Ohm's law fails, but the Einstein relation (suitably interpreted) holds. We also apply our results to another relevant model - Galton board.