A representation of the moment measures of the general ideal Bose gas
type: Item
instance of: scholarly article
zbMATH Open document ID: 1273.35159
published in: Mathematische Nachrichten
publication date: 2012-06-04T00:00:00Z
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 6042721
zbMATH Keywords: infinitely divisible point process
zbMATH Keywords: integration by parts formula
zbMATH Keywords: random KMM measure
zbMATH Keywords: general ideal Bose gas
author: Benjamin Nehring
author: Hans Zessin
MaRDI profile type: Publication
full work available at URL: mana.201000111
OpenAlex ID: W1972151840
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DOI: 10.1002/MANA.201000111
title: A representation of the moment measures of the general ideal Bose gas
review text: The aim of the paper is to exhibit the permanental structure of the ideal Bose gas, by employing a new approach which combines a characterization of infinitely divisible random measures with a decomposition of the moment measures into its factorial measures. This is a reformulation of the famous fundamental work of \textit{K.-H. Fichtner} [Math. Nachr. 151, 59--67 (1991; Zbl 0728.60052)]. As a matter of fact, the authors exhibit the moment measures of all orders of the general ideal Bose gas in terms of loop integrals. This representation is nothing but a point process analogue of the old idea of Symanzik from 1969 for Euclidean quantum field theory.NEWLINENEWLINEWe can characterize infinitely divisible random measures on a general state space from the point of view of its Cambell measure. It is shown that such random measures are characterized by some integration by parts formula. Their proof is due to a direct approach, and its leading philosophy can be found in the seminal work of \textit{J. Mecke} [Z.\ Wahrscheinlichkeitstheor.\ Verw.\ Geb.\ 9, 36--58 (1967; Zbl 0164.46601)].NEWLINENEWLINETheorem 1. Let \(\alpha\) be an element in the space \({\mathcal M}(X)\) of locally finite measures on a Polish state space \(X\), and let \(L\) be a measure on \(Y= {\mathcal M}(X) \setminus \{0 \}\) of first order. Then the following conditions are equivalent for a random measure \(\tilde{\xi}\): \newline (a) \(\tilde{\xi}\) is infinitely divisible for \(( \alpha, L)\); \newline (b) the Laplace transform of \(\tilde{\xi}\) is given by NEWLINE\[NEWLINE {\mathcal L}_P(f) = \exp \{ - ( \alpha(f) + L( 1 - \exp\{ - \zeta_f \}) ) \} \qquad \text{for} \quad f \in F_+(X), \tag{1} NEWLINE\]NEWLINE where \(F_+(X)\) denotes the collection of measurable, non-negative numerical functions, and \(\nu(f)\) is an integral of \(f\) with respect to a measure \(\nu\) and \(\zeta_f\) : \(\mu \mapsto \mu(f)\) is defined on \(Y\) \(=\) \({\mathcal M}(X) \setminus \{ 0 \}\); \newline (c) \(\tilde{\x…
reviewed by: Isamu Dôku
description: scientific article; zbMATH DE number 6042721