Positive operators in statistical mechanics of continuous system
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 \(\rho <[(e + 1)C(\beta)]^{-1}\).
scientific article; zbMATH DE number 5083442