Gradient Dynamics of Infinite Point Systems

Abstract

Nonequilibrium gradient dynamics of d-dimensional particle systems is investigated. The interaction is given by a superstable pair potential of finite range. Solutions are constructed in the well-defined set of locally finite configurations with a logarithmic order of energy fluctuations. If the system is deterministic and d < 2, then singular potentials are also allowed. For stochastic models with a smooth interaction we need d < 4. In order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium, we investigate smoothness of the Markov semigroup and describe some properties of its generator. 0. Introduction. The purpose of this paper is to study existence and regularity properties of solutions to the following infinite system of stochastic differential equations. Consider a countable set S of d-dimensional particles suspended in a fluid, where the interaction is given by a pair potential U: Rgd (- 00, + oo]. In a quasi-microscopic description of such systems the effect of collisions with the particles of the fluid can be represented by uncorrelated stochastic forces, and the soft resistance of the liquid medium reduces the order of the equations of motion from two to one, see [13], [23], and [24] for some further references. Configurations of these systems are countable subsets of Rd such that any bounded domain contains a finite number of points only. Particles of a configuration will be identified by labelling points by elements of S. Thus a