Scaling limit of stochastic dynamics in classical continuous systems

We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on ${\mathbb R}^d, d \ge 1$. The aim is to derive macroscopic quantities from a given micro- or mesoscopic system. The scaling we consider has been investigated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}, and \cite{GP86}, 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 in \cite{AKR98a}, \cite{AKR98b}, 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 e.g. 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 in \cite{Br80}, \cite{Ro81}, \cite{Sp86}

Kawasaki dynamics in the continuum via generating functionals evolution

We construct the time evolution of Kawasaki dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, which leads to a local (in time) solution. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.Comment: 13 page

Glauber dynamics in the continuum via generating functionals evolution

We construct the time evolution for states of Glauber dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, leading to a local (in time) solution which, under certain initial conditions, might be extended to a global one. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.Comment: 24 page