N/V-limit for stochastic dynamics in continuous particle systems. Probab. Theory Relat

Abstract. We provide an N=V-limit for the innite particle, innite volume stochas-tic dynamics associated with Gibbs states in continuous particle systems on Rd; d 1. Starting point is an N-particle stochastic dynamic with singular interaction and re-ecting boundary condition in a subset Rd with nite volume (Lebesgue measure) V = jj <1. The aim is to approximate the innite particle, innite volume stochas-tic dynamic by the above N-particle dynamic in as N! 1 and V!1 such that N=V! , where is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons{Zheng decomposition. The equilibrium measures of the accumulation points are identied as innite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identied as innite particle, innite volume stochastic dynamics via the associ-ated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials of Ruelle type and all temper-atures, densities, and dimensions d 1, respectively. may have a nontrivial negative part and innite range as e.g. the Lennard{Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by nite volume canonical Gibbs measures with empty boundary condition. 1