Gradient dynamics of infinite point systems
A countable set S of d-dimensional particles suspended in a fluid is considered, where the interaction is given by a pair potential \(U: {\mathbb{R}}^ d\to (-\infty,\infty]\) with finite range. The model is represented by the infinite system of stochastic differential equations \[ d\omega_ k=-\sum_{j\neq k}\text{grad} U(\omega_ k-\omega_ j)dt+\sigma dw_ k,\quad k\in S, \] where \((\omega_ k)_{k\in S}\) is the configuration of the particle system, \((w_ k)_{k\in S}\) is a family of independent standard Wiener processes in \({\mathbb{R}}^ d\), and \(\sigma\geq 0\) is a constant. The purpose of the paper is to study existence and regularity properties of solutions of this system of equations. For smooth interactions, existence and uniqueness of nonequilibrium solutions is proved if \(d\leq 4\), and for a singular U if \(d\leq 2\) and \(\sigma =0\). Solutions are constructed as a.s. limits of solutions of finite subsystems. Properties of the Markov semigroups are investigated in order to develop prerequisites for the theory of hydrodynamical fluctuations in equilibrium.
infinite system of stochastic differential equations
existence and uniqueness of nonequilibrium solutions
hydrodynamical fluctuations in equilibrium
scientific article; zbMATH DE number 4011629