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Equilibrium Kawasaki dynamics of continuous particle systems
 INFIN. DIMENS. ANAL. QUANTUM PROBAB. RELAT. TOP
, 2007
"... We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a prior ..."
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Cited by 8 (2 self)
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We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birthanddeath process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).
Infinite interaction diffusion particles I: Equilibrium process and its scaling limit
 Forum Math
"... A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y) ..."
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Cited by 6 (3 self)
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A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics—the socalled gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form EΓ µ on L2 (Γ; µ), and under general conditions on the potential φ, prove its closability. For a potential φ having a “weak ” singularity at zero, we also write down an explicit form of the generator of EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′), where D ′ is the dual space of D:=C ∞ 0 (Rd).
N/VLIMIT FOR STOCHASTIC DYNAMICS IN CONTINUOUS PARTICLE SYSTEMS
, 2005
"... We provide an N/Vlimit 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 Nparticle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ ⊂ R d with f ..."
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Cited by 3 (3 self)
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We provide an N/Vlimit 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 Nparticle 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 Nparticle 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 selfadjointness, 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 byproduct an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.
On convergence of dynamics of hopping particles to a birthanddeath process in continuum
, 2008
"... We show that some classes of birthanddeath processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics) ..."
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We show that some classes of birthanddeath processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics)
Diffusion approximation for equilibrium Kawasaki dynamics in continuum
, 2007
"... A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derive ..."
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A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, φ, (in particular, admitting a singularity of φ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finitedimensional distributions of the corresponding equilibrium processes. In particular, if the potential φ is from C 3 b (Rd) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi et al., J. Math.