2009/01/01 by N. Chernov, Dmitry Dolgopyat, D. Dolgopyat · 76 citations
Economics, Econometrics and Finance · Mathematics · #Brownian motion #Center of mass (relativistic) #Chaotic #Classical mechanics #Complex Systems and Time Series Analysis #Container (type theory) #Geometry #Mathematical Dynamics and Fractals #Mathematics #Mechanics #Particle (ecology) #Physics #Position (finance) #Quantum mechanics #Scaling #Scaling limit #Stochastic processes and statistical mechanics #Work (physics)
paper · doi:10.1090/memo/0927
published in Memoirs of the American Mathematical Society 198(927), 0 (American Mathematical Society)
openalex publication_date 2009/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
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 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 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.