N/V-limit for stochastic dynamics in continuous particle systems |
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Authors: | Martin Grothaus Yuri G Kondratiev Michael Röckner |
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Institution: | (1) Mathematics Department, University of Kaiserslautern, P.O.Box 3049, 67653 Kaiserslautern, Germany;(2) BiBoS, Bielefeld University, 33615 Bielefeld, Germany;(3) SFB 611, IAM, University of Bonn, 53115 Bonn, Germany;(4) BiBoS and Mathematics Department, Bielefeld University, 33615 Bielefeld, Germany;(5) Inst. Math., NASU, 252601 Kiev, Ukraine |
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Abstract: | We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle
systems on ℝ
d
,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ
d
with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ 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
identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves
are identified as infinite particle, infinite volume stochastic dynamics via the associated 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 ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides
as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary
condition. |
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Keywords: | 60B12 82C22 60K35 60J60 60H10 |
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