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Brownian Brownian Motion-I

Nikolai ChernovDmitry Dolgopyat

出版	American Mathematical Soc., 2009-03-06
主題	Science / Physics / Mathematical & Computational
ISBN	082184282X9780821842829
URL	http://books.google.com.hk/books?id=oRq9AwAAQBAJ&hl=&source=gbs_api
EBook	SAMPLE

註釋

A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work the authors study a 2D version of this model, where the molecule is a heavy disk of mass $M \gg 1$ and the gas is represented by just one point particle of mass $m=1$, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of the particles is ensured by convex (scattering) walls of the container. The authors prove that the position and velocity of the disk, in an appropriate time scale, converge, as $M\to\infty$, to a Brownian motion (possibly, inhomogeneous); the scaling regime and the structure of the limit process depend on the initial conditions. The proofs are based on strong hyperbolicity of the underlying dynamics, fast decay of correlations in systems with elastic collisions (billiards), and methods of averaging theory.