FROM DISCRETE DETERMINISTIC DYNAMICS TO BROWNIAN MOTIONS

Abstract

Two types of point particles, large and small, with state space ℝd are considered, where d ≥ 2 and "large" and "small" refer to different masses. The small particles move deterministically with very large initial velocities. They transfer their momenta to the large particles through a smooth mean field interaction which completely determines the motion of the large particles. The joint dynamics is described in a spacetime lattice by an Euler scheme for coupled oscillators with a friction term for the large particles. This lattice defines the mesoscale for the system. A scaling limit leads to a replacement of the mesoscale by the macroscale as follows: The very large initial velocities are assumed to be independent and they let a small particle interact with a large particle only for a short time, after which the particle escapes to infinity and new particles start interacting with the large particles. Thus, the initial spatial independence of the small particles causes independence in the time increments of the velocities of the large particles. Therefore, as the friction of the large particles and the speed of the small particles tend to infinity in this scaling limit, the positions of the large particles become the positions of spatially correlated Brownian motions whose correlations can be computed from the interaction force. A similar result holds for a system without friction, where the velocities of the large particles become spatially correlated Brownian motions.