Deterministic Dynamics in a Lattice Model and a Mesoscopic (Stochastic) Limit

The evolution of a space—time discrete version of the Newtonian system (1.4) is analyzed on a fixed (macroscopic) time interval [0, ̂t] (cf. (2.9)). The interaction between large and small particles is governed by a twice continuously differentiable odd R^d-valued function G.¹ We assume that all partial derivatives up to order 2 are square integrable and that ǀGǀ^m is integrable for 1 ⩽ m ⩽ 4, where “integrable” refers to the Lebesgue measure on R^d. The function G will be approximated by odd R^d-valued functions G _n with bounded supports (cf. (2.1)). Existence of the space—time discrete version of (1.4) is derived employing coarse graining in space and an Euler scheme in time. The mesoscopic limit (2.11) is a system stochastic ordinary differential equation (SODEs) for the positions of the large particles. The SODEs are driven by Gaussian standard space—time white noise that may be interpreted as a limiting centered number density of the small particles. The proof of the mesoscopic limit theorem (Theorem 2.4) is provided in Chap. 3.