Positive operators in statistical mechanics of continuous system
type: Item
instance of: scholarly article
title: Positive operators in statistical mechanics of continuous system
zbMATH Open document ID: 1108.82008
author: Marek Gorzelańczyk
published in: Journal of Functional Analysis
publication date: 2007-01-09T00:00:00Z
review text: It is known that to establish the phase transition in the statistical mechanics one needs to investigate the analytical properties of the Kirkwood-Salsburg (K-S) integral equation. In the present paper the Kirkwood-Salsburg equation is written as an operator. In the standard approach [see for example, \textit{D. Y. Petrina, V. I. Gerasimenko} and \textit{P. V. Malyshev}, Mathematical foundation of classical statistical mechanics, Advanced Studies in Contemporary Mathematics Series, 6. New York etc.: Gordon and Breach Science Publishers (1989; Zbl 0728.46053)] the main object was the estimation of the K-S operator spectral radius by means of its norm in the complex Banach space. Here in the paper, the (K-S) equations are studied in the real Banach space. That is why, the author applied the theory of positive operators (cone preserving operators) in the case of the positive potentials. It is established that the (K-S) operators and their conjugate operators are positive in the case of any arbitrary volume. This observation gives a new information on localization of the spectrum of the (K-S) operator and the zeroes of the grand canonical partition function. It is found that for much larger values of chemical activity \(z\) and density \(\rho\), as it is known, such phase transitions do not occur. The analyticity on \(z < [(e-2)C(\beta)]^{-1}\) of the pressure and one particle correlation functions are shown. It is obtained that a phase transition does not occur for the density \( \rho <[(e-1)C(\beta)]^{-1} \). This is essentially a larger region of analyticity than the previously known \(\rho <[(e + 1)C(\beta)]^{-1}\).
reviewed by: Farruh Mukhamedov
Mathematics Subject Classification ID: 82B21
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47N55
Mathematics Subject Classification ID: 82B26
zbMATH DE Number: 5083442
zbMATH Keywords: positive operators
zbMATH Keywords: cones in real Banach space
zbMATH Keywords: Kirkwood-Salsburg integral equation
zbMATH Keywords: phase transition
MaRDI profile type: Publication
full work available at URL: j.jfa.2006.04.018
OpenAlex ID: W1969045694
cites work: Equilibrium classical statistical mechanics of continuous systems in the Kirkwood-Salsburg approach
cites work: Q3992446
cites work: Q3999260
cites work: Some remarks on the ground state of infinite systems in statistical mechanics
DOI: 10.1016/J.JFA.2006.04.018
description: scientific article; zbMATH DE number 5083442