Penrose-Stable Interactions in Classical Statistical Mechanics

For a pair potential Φ in a general underlying space X satisfying some natural and sufficiently general conditions in the sense of Penrose (J Math Phys 4:1312, 1963) and Poghosyan and Ueltschi (J Math Phys 50:053509, 2009) together with a locally finite measure ϱ on X we define by means of the so-called Ursell kernel a function r which is shown to be the correlation function of a unique process G , the limiting Gibbs process for (Φ ,ϱ ) with empty boundary conditions. This process is exhibited as a Gibbs process in the sense of Dobrushin, Lanford and Ruelle for a class of pair potentials, which contains classical stable and hard-core potentials that are called Penrose potentials here. Particularly, a class of positive potentials is included. Finally, for some class of Penrose potentials, we show that G is the unique Gibbs process for Φ . We use the classical method of Kirkwood–Salsburg equations. A decisive role is played by a generalization of Ruelle’s estimate for correlation functions.