Scaling limit of stochastic dynamics in classical continuous systems

• 成果类型: Article
• 署名作者: Grothaus, M; Kondratiev, YG; Lytvynov, E; Röckner, M
• 署名单位: University of Bielefeld; University of Bonn; University of Bielefeld; National Academy of Sciences Ukraine; Institute of Mathematics of NASU
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1494-1532
• 关键词: dimensional wiener-processes; interacting diffusion-processes; statistical mechanical systems; CONFIGURATION-SPACES; dirichlet forms; brownian balls; particles; CONSTRUCTION; uniqueness; geometry

摘要

We investigate a scaling limit of gradient Stochastic dynamics associated with Gibbs states in classical continuous systems on R-d, d greater than or equal to 1. The aim is to derive macroscopic quantities from a given microscopic or mesoscopic system. The scaling we consider has been investigated by Brox (in 1980), Rost (in 1981), Spohn (in 1986) and Guo and Papanicolaou (in 1985), under the assumption that the underlying potential is in C-0(3) and positive. We prove that the Dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the Dirichlet form of a generalized Ornstein-Uhlenbeck process. The proof is based on the analysis and geometry on the configuration space which was developed by Albeverio, Kondratiev and Rockner (in 1998), and works for general Gibbs measures of Ruelle type. Hence, the underlying potential may have a singularity at the origin, only has to be bounded from below and may not be compactly supported. Therefore, singular interactions of physical interest are covered, as, for example, the one given by the Lennard-Jones potential, which is studied in the theory of fluids. Furthermore, using the Lyons-Zheng decomposition we give a simple proof for the tightness of the scaled processes. We also prove that the corresponding generators, however, do not converge in the L-2-sense. This settles a conjecture formulated by Brox, by Rost and by Spohn.