Non-Linear Generalization of the DLR Equations: -Specifications and -Equilibrium Measures
arXiv:2601.06470
Abstract
We introduce a {\it non-linear} generalization of the classical Dobrushin-Lanford-Ruelle (DLR) framework by developing the concept of a -specification and the associated -equilibrium measures. These objects arise naturally from a family of non-linear -stochastic operators acting on the space of probability measures. A -equilibrium measure is characterized as a fixed point of such operators, providing a non-linear analogue of the Gibbs equilibrium in the sense of DLR. We establish general conditions ensuring the existence and uniqueness of -equilibrium measures and demonstrate how quasilocality plays a decisive role in their construction. Moreover, we exhibit examples of -specifications with an empty set of -equilibrium measures. We characterize the set of -equilibrium measures by studying the dynamical systems generated by a class of -stochastic operators. As a concrete application, we show that for the one-dimensional Ising model at sufficiently low temperatures, multiple -equilibrium measures may exist, even though the classical Gibbs measure remains unique. Our results reveal that the -specification formalism extends the DLR theory from linear to non-linear settings and opens a new direction in the study of Gibbs measures and equilibrium states of physical systems.
25 pages