Construction of point processes for classical and quantum gases

Nehring, Benjamin

doi:10.1063/1.4807080

Cited by 3 publications

(10 citation statements)

References 20 publications

(23 reference statements)

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“…(f ∈ F ) Recall that E φ is defined in (17). This is in general a signed measure on the Polish space X of finite configurations in the phase space.…”

Section: To Motivate the Main Results In Section 5 We Now Present Inmentioning

confidence: 99%

“…The convergence of the cle-method has been shown in [17] in the following precise sense: Under the conditions (A 1 ) and (A 2 ) the sequence of processes Q G , G ∈ B 0 (X), converges weakly, as G ↑ X, to some point process L having Lévy-measure L. Recall that this terminology means that the Laplace transform of L is of the form K L . The process L is called here the KMMprocess with Lévy measure L. Moreover, the process L solves the following equation:…”

Section: The Cle-methods In the Infinitely Extended Casementioning

confidence: 99%

“…Then we indicate briefly the infinitely extended case. This construction has been developed in [17,18]. …”

Section: A General Construction Of Processes By Means Of the Cluster mentioning

confidence: 99%

“…[17]) that the corresponding process determining measures are given by det (K(a i , a j ) i,j ) λ(d a 1 ) . .…”

Section: Permanental and Determinantal Processes And Its Modificationsmentioning

confidence: 99%

“…The method we use is a new version of cluster expansions which had been developed in [18,17] and which is summarized in Theorem 1. In a first step we then construct in Theorem 2, in the context of statistical mechanics, by means of this method limiting interacting processes by combining a recent result of Poghosyan and Ueltschi [21] with Theorem 1 .…”

Section: Introductory Remarksmentioning

confidence: 99%

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On the construction of point processes in statistical mechanics

Nehring

Poghosyan

Zessin

2013

Journal of Mathematical Physics

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We present a new approach to the construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd of Ginibre's Fermi-Dirac gas of such loops. This approach is based on the cluster expansion method. We obtain the existence of Gibbs perturbations of a large class of point processes. Moreover, it is shown that certain “limiting Gibbs processes” are Gibbs in the sense of Dobrushin, Lanford, and Ruelle if the underlying potential is positive. Finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.

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“…(f ∈ F ) Recall that E φ is defined in (17). This is in general a signed measure on the Polish space X of finite configurations in the phase space.…”

Section: To Motivate the Main Results In Section 5 We Now Present Inmentioning

confidence: 99%

Section: The Cle-methods In the Infinitely Extended Casementioning

confidence: 99%

“…Then we indicate briefly the infinitely extended case. This construction has been developed in [17,18]. …”

Section: A General Construction Of Processes By Means Of the Cluster mentioning

confidence: 99%

“…[17]) that the corresponding process determining measures are given by det (K(a i , a j ) i,j ) λ(d a 1 ) . .…”

Section: Permanental and Determinantal Processes And Its Modificationsmentioning

confidence: 99%

Section: Introductory Remarksmentioning

confidence: 99%

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On the construction of point processes in statistical mechanics

Nehring

Poghosyan

Zessin

2013

Journal of Mathematical Physics

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Splitting‐characterizations of the Papangelou process

Nehring

Rafler

Zessin

2015

Mathematische Nachrichten

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For point processes we establish a link between integration-by-parts-and splitting-formulas which can also be considered as integration-by-parts-formulas of a new type. First we characterize finite Papangelou processes in terms of their splitting kernels. The main part then consists in extending these results to the case of infinitely extended Papangelou and, in particular, Pólya and Gibbs processes. and denoted also by P ε z, . Here ε ∈ {−1, +1}. This class of processes has its origin in [26]. Finally the class of Gibbs processes G(V, ) is of fundamental importance. Such processes are, in general not uniquely, specified by some reference measure ∈ M and some abstract Boltzmann factor V (x, μ) by which one understands a non-negative, measurable function having the property V (x, ν + κ) = V (x, ν)V (x, κ), x ∈ X, ν, κ ∈ M ·· .

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A central limit theorem for a classical gas

Zessin,

Poghosyan

2023

Adv Cont Discr Mod

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For a class of translation-invariant pair potentials ϕ in $(\mathbb{R}^{d},z\lambda )$ ( R d , z λ ) satisfying a stability and regularity condition, we choose z so small that the associated collection $\mathcal{ G}(\phi,z\lambda )$ G ( ϕ , z λ ) of Gibbs processes contains at least the stationary process G, which is a Gibbs process in the sense of DLR and is given by the limiting Gibbs process with empty boundary conditions. Using an abstract version of the method of cluster expansions and Dobrushin’s approach to the central limit theorem, we present a central limit theorem for the particle numbers of G.

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Construction of point processes for classical and quantum gases

Cited by 3 publications

References 20 publications

On the construction of point processes in statistical mechanics

On the construction of point processes in statistical mechanics

Splitting‐characterizations of the Papangelou process

A central limit theorem for a classical gas

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