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Non-Linear Generalization of the DLR Equations: $q$-Specifications and $q$-Equilibrium Measures

F. H. Haydarov, B. A. Omirov, U. A. Rozikov

TL;DR

This work extends Gibbsian equilibrium theory by introducing a nonlinear generalization of DLR through $q$-specifications and $q$-equilibrium measures, defined as fixed points of nonlinear $q$-stochastic operators. It develops the foundations (kernels, specifications, and quasilocality), proves existence and analyzes uniqueness via boundary-condition sensitivity, and constructs examples including an empty-equilibrium case. A dynamical-systems perspective is used to link fixed points to the long-term behavior of $q$-SOs, and the framework is applied to the 1D Ising model to show the emergence of multiple $q$-equilibria at low temperature, contrasting with the classical Gibbs uniqueness. The results reveal a rich nonlinear extension of DLR theory, with potential implications for non-additive entropy and nonlinear interacting systems across physics and related fields.

Abstract

We introduce a {\it non-linear} generalization of the classical Dobrushin-Lanford-Ruelle (DLR) framework by developing the concept of a $q$-specification and the associated $q$-equilibrium measures. These objects arise naturally from a family of non-linear $q$-stochastic operators acting on the space of probability measures. A $q$-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 $q$-equilibrium measures and demonstrate how quasilocality plays a decisive role in their construction. Moreover, we exhibit examples of $q$-specifications with an empty set of $q$-equilibrium measures. We characterize the set of $q$-equilibrium measures by studying the dynamical systems generated by a class of $q$-stochastic operators. As a concrete application, we show that for the one-dimensional Ising model at sufficiently low temperatures, multiple $q$-equilibrium measures may exist, even though the classical Gibbs measure remains unique. Our results reveal that the $q$-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.

Non-Linear Generalization of the DLR Equations: $q$-Specifications and $q$-Equilibrium Measures

TL;DR

This work extends Gibbsian equilibrium theory by introducing a nonlinear generalization of DLR through -specifications and -equilibrium measures, defined as fixed points of nonlinear -stochastic operators. It develops the foundations (kernels, specifications, and quasilocality), proves existence and analyzes uniqueness via boundary-condition sensitivity, and constructs examples including an empty-equilibrium case. A dynamical-systems perspective is used to link fixed points to the long-term behavior of -SOs, and the framework is applied to the 1D Ising model to show the emergence of multiple -equilibria at low temperature, contrasting with the classical Gibbs uniqueness. The results reveal a rich nonlinear extension of DLR theory, with potential implications for non-additive entropy and nonlinear interacting systems across physics and related fields.

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.
Paper Structure (14 sections, 17 theorems, 152 equations)

This paper contains 14 sections, 17 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Setup
  4. Stochastic operators
  5. $q$-Stochastic operators and their connections to known transformations
  6. Gibbs Measures
  7. $q$-equilibrium measures
  8. Probability $q$-Kernels and $q$-Specifications
  9. Gibbs $q$-specification
  10. Existence of $q$-equilibrium measure
  11. Uniqueness via sensitivity to boundary conditions
  12. A $q$-specification with empty-set of $q$-equilibrium measures
  13. Dynamical systems generated by $q$-SO and $q$-equilibrium measures
  14. 1D Ising model with multiple $q$-equilibrium measures

Key Result

Lemma 1

The $q$-kernel is idempotent, i.e., $\gamma_\Lambda\gamma_\Lambda=\gamma_\Lambda$, for any $\Lambda\Subset V$.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 4
  • Lemma 3
  • ...and 34 more