Abstract
In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood–Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labeled by certain forest graphs, the connected components of which are tree graphs. We generalize the method developed by Minlos and Poghosyan [Theor. Math. Phys. 31, 408 (1977)] in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF, which were obtained earlier for lattice spin systems.
Highlights
Correlation functions were first introduced in statistical mechanics by Ornstein and Zernike at the beginning of the 20th century in the study of critical fluctuations; see Ref. 1
The physical correlations between particles are described by the socalled truncated correlation functions (TCF) or connected correlation functions, which become zero in the absence of interaction between the particles
In Ref. 9, Lebowitz derived bounds on the decay of correlations between two widely separated sets of particles for ferromagnetic Ising spin systems in terms of the decay of the pair correlation
Summary
Correlation functions were first introduced in statistical mechanics by Ornstein and Zernike at the beginning of the 20th century in the study of critical fluctuations; see Ref. 1. In Ref. 13, some general results on strong cluster properties of TCF and PTCF for lattice gases are presented (the proof of their main theorem involves long technical parts, which were obtained in unpublished work of one of the authors). We derive equations of Kirkwood–Salsburg-type for the PTCF and apply the technique that was proposed by Minlos and Poghosyan in Ref. 14 to obtain solutions of these equations in the form of a series of contributions of certain forest diagrams. Such a representation makes it possible to obtain strong cluster properties for the PTCF in a convenient form for deriving estimates. We stress the point that explicit formulas for the upper bounds are obtained, some of which rely on some original (to our best knowledge) combinatorial identities
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