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N/V-limit for stochastic dynamics in continuous particle systems
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  • Published: 24 April 2006

N/V-limit for stochastic dynamics in continuous particle systems

  • Martin Grothaus1,2,3,
  • Yuri G. Kondratiev4,5 &
  • Michael Röckner4 

Probability Theory and Related Fields volume 137, pages 121–160 (2007)Cite this article

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  • 8 Citations

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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 ℝd,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝd 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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Authors and Affiliations

  1. Mathematics Department, University of Kaiserslautern, P.O.Box 3049, 67653, Kaiserslautern, Germany

    Martin Grothaus

  2. BiBoS, Bielefeld University, 33615, Bielefeld, Germany

    Martin Grothaus

  3. SFB 611, IAM, University of Bonn, 53115, Bonn, Germany

    Martin Grothaus

  4. BiBoS and Mathematics Department, Bielefeld University, 33615, Bielefeld, Germany

    Yuri G. Kondratiev & Michael Röckner

  5. Inst. Math., NASU, 252601, Kiev, Ukraine

    Yuri G. Kondratiev

Authors
  1. Martin Grothaus
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  3. Michael Röckner
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Correspondence to Martin Grothaus.

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Grothaus, M., Kondratiev, Y. & Röckner, M. N/V-limit for stochastic dynamics in continuous particle systems. Probab. Theory Relat. Fields 137, 121–160 (2007). https://doi.org/10.1007/s00440-006-0499-y

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  • Received: 26 September 2005

  • Published: 24 April 2006

  • Issue Date: January 2007

  • DOI: https://doi.org/10.1007/s00440-006-0499-y

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Mathematics Subject Classification (2000)

  • 60B12
  • 82C22
  • 60K35
  • 60J60
  • 60H10

Key words and phrases

  • Limit theorems
  • Interacting particle systems
  • Diffusion processes
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