2014/04/11 by Leandro Cioletti, Cioletti, Leandro, Artur O. Lopes +1
Mathematics · Physics and Astronomy · #28Dxx #37D35 (Primary) #82B05 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Statistical Mechanics and Entropy #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1404.3232
openalex publication_date 2014/04/11 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this paper, we describe several different meanings for the concept of\nGibbs measure on the lattice \ℕ in the context of finite alphabets\n(or state space). We compare and analyze these "in principle" distinct notions:\nDLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of\nthe Ruelle operator (also called conformal measures).\n Among other things we extended the classical notion of a Gibbsian\nspecification on \ℕ in such way that the similarity of many results\nin Statistical Mechanics and Dynamical System becomes apparent. One of our main\nresult claims that the construction of the conformal Measures in Dynamical\nSystems for Walters potentials, using the Ruelle operator, can be formulated in\nterms of Specification. We also describe the Ising model, with\n1/r2+\ε interaction energy, in the Thermodynamic Formalism\nsetting and prove that its associated potential is in Walters space - we\npresent an explicit expression. We also provide an alternative way for\nobtaining the uniqueness of the DLR-Gibbs measures.\n