| eBook ISBN: | 978-1-4704-0533-5 |
| Product Code: | MEMO/198/927.E |
| List Price: | $82.00 |
| MAA Member Price: | $73.80 |
| AMS Member Price: | $49.20 |
| eBook ISBN: | 978-1-4704-0533-5 |
| Product Code: | MEMO/198/927.E |
| List Price: | $82.00 |
| MAA Member Price: | $73.80 |
| AMS Member Price: | $49.20 |
-
Book DetailsMemoirs of the American Mathematical SocietyVolume: 198; 2009; 193 ppMSC: Primary 37; Secondary 34; 60
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.
-
Table of Contents
-
Chapters
-
Chapter 1. Introduction
-
Chapter 2. Statement of results
-
Chapter 3. Plan of the proofs
-
Chapter 4. Standard pairs and equidistribution
-
Chapter 5. Regularity of the diffusion matrix
-
Chapter 6. Moment estimates
-
Chapter 7. Fast slow particle
-
Chapter 8. Small large particle
-
Chapter 9. Open problems
-
Appendix A. Statistical properties of dispersing billiards
-
Appendix B. Growth and distortion in dispersing billiards
-
Appendix C. Distortion bounds for two particle system
-
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Requests
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.
-
Chapters
-
Chapter 1. Introduction
-
Chapter 2. Statement of results
-
Chapter 3. Plan of the proofs
-
Chapter 4. Standard pairs and equidistribution
-
Chapter 5. Regularity of the diffusion matrix
-
Chapter 6. Moment estimates
-
Chapter 7. Fast slow particle
-
Chapter 8. Small large particle
-
Chapter 9. Open problems
-
Appendix A. Statistical properties of dispersing billiards
-
Appendix B. Growth and distortion in dispersing billiards
-
Appendix C. Distortion bounds for two particle system
