Anomalous Diffusion in Deterministic Systems

点击翻译按钮获取中文摘要

Certain gas models (planar periodic Lorentz flows) were introduced as deterministic models for Brownian motion. A statistical law called the almost sure invariance principle (ASIP) makes precise the connection with Brownian motion. Recent work established the ASIP for the Lorentz flow.Brownian motion has a characteristic growth rate of order sqrt t (square root of t ) where t denotes time. Currently, there is a great deal of interest in anomalous diffusion with a different growth rate. The aim of this project is the systematic study of anomalous diffusion (both subdiffusive and superdiffusive) in deterministic systems. In the superdiffusive (fast) case, this means proving the ASIP with Brownian motion replaced by a Levy process. In the subdiffusive (slow) case, we will focus on Sinai diffusion where the growth rate is of order (log t)^2.The results will apply to infinite horizon Lorentz gases and Bunimovich stadia. In addition, there are applications in dynamical systems with symmetry, including spiral waves in excitable media, and travelling waves.