Scaling limit of stochastic dynamics in classical continuous systems

Abstract

We investigate a scaling limit of gradient stochastic dynamics associated with Gibbs states in classical continuous systems on ${\mathbb R}^d$, $d \ge 1$. 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 $C^3_0$ 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 Röckner (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.