12/13/2023 0 Comments Entropy of subshiftIn the complement of these regions, the configurations have to be homogeneous and of minimal energy. ![]() Definitions and NotationsĪs mentioned in the introduction, the Häggström correspondence takes advantage of the fact that for some models of equilibrium statistical mechanics, the energy is concentrated on some lattice regions that we refer to as contours. In Section 4 and Section 5, we illustrate this correspondence in the case of the Potts and the six-vertex model, respectively. In Section 3, we develop the SFT–SMM correspondence. The rest of the paper is organized as follows: in Section 2, we introduce some basic notions that we use throughout the paper. In order to illustrate the aforementioned correspondence and the nature of the applications we invoke, we use the Potts model and the six-vertex model of statistical mechanics. Although the construction can be carried out in any dimension, for the sake of concreteness, we will restrict ourselves to dimension two, where the relevant phenomenology already appears. Our construction also makes explicit the correspondence between the parametrization of the family of SFTs and the inverse temperature in the corresponding SMM, making sense of a phase transition in the symbolic context. Ours is simpler in what concerns the construction the subshift of finite type, as well in what concerns the proof of the equivalence. The correspondence we study in this paper is a simplified version of the one due to O. In this paper, we revisit this correspondence with the aim of pointing out further applications of statistical mechanics results to symbolic dynamics. In, Häggström formalizes and generalizes the above-mentioned correspondence in such a way that for each equilibrium state of the statistical mechanics model (SMM), there is a measure of maximal entropy for the corresponding subshift of finite type. Zahradnik in, a complete description of the simplex of measures of maximal entropy. Using the same strategy, Burton and Steif derive in, using an idea analogous to the one used by M. Indeed, one of the results obtained in, the existence of a strongly irreducible subshift of finite type in dimension, with two supporting at least two ergodic measures of maximal entropy, is the translation of a result by Peierls concerning the Ising model. The success of this approach lies in the fact that it furnishes a dictionary between equilibrium statistical mechanics and symbolic dynamics, translating rigorous results from statistical mechanics to symbolic dynamics. Häggström in, consists on making subshifts of finite type correspond to statistical mechanics models, in such a way that equilibrium states for the statistical mechanics model correspond to measures of maximal entropy for the symbolic system. Steif developed a strategy to construct examples of transitive subshifts of finite type admitting several measures of maximal entropy. In higher dimensions, transitivity is not enough to ensure that a subshift of finite type is intrinsically ergodic. A topological dynamical system with a unique measure of maximal entropy is qualified as intrinsically ergodic. For transitive one-dimensional subshifts of finite type on a finite number of states, there exists one and only one invariant measure achieving the topological entropy, which, on the other hand, is the supremum of the metric entropies (see for instance). ![]() A subshift of finite type is a symbolic dynamical system determined by a finite collection of forbidden patterns.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |