On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection

Abstract

We consider $(M,d)$ a connected and compact manifold and we denote by ${\mathcal{B}}_i$ the Bernoulli space $M^{\mathbb{Z}}$ of sequences represented by $$x=(...x_{-3},x_{-2},x_{-1},x_0,x_1,x_2,x_3,...),$$ where $x_i$ belongs to the space (alphabet) $M$. The case where $M={\mathbb{S}}^1$, the unit circle, is of particular interest here. The analogous problem in the one-dimensional lattice $\mathbb{N}$ is also considered. %In this case we consider the potential $A:\mathcal{B}=M^{\mathbb{N}}\to \mathbb{R}.$ Let $A: \mathcal{B}_i \rar \R$ be an {\it observable} or {\it potential} defined in the Bernoulli space ${\mathcal{B}}_i$. The potential $A$ describes an interaction between sites in the one-dimensional lattice $M^{\mathbb{Z}}$. Given a temperature $T$, we analyze the main properties of the Gibbs state ${\hat{\mu }}_{\frac{1}{T}A}$ which is a certain probability measure over ${\mathcal{B}}_i$. We denote this setting "the general XY model". In order to do our analysis we consider the Ruelle operator associated to $\frac{1}{T}A$, and, we get in this procedure the main eigenfunction ${\psi }_{\frac{1}{T}A}$. Later, we analyze selection problems when temperature goes to zero: a) existence, or not, of the limit (on the uniform convergence) $$V:=\underset{T\to 0}{\lim }T\log ({\psi }_{\frac{1}{T}A}),\text{a question about selection of subaction},$$ and, b) existence, or not, of the limit (on the weak${}^{\ast }$ sense) $$\tilde{\mu }:=\underset{T\to 0}{\lim }{\hat{\mu }}_{\frac{1}{T}A},\text{a question about selection of measure}.$$ The existence of subactions and other properties of Ergodic Optimization are also considered.

References in corpus (2)

• On the zero-temperature limit of Gibbs states
• Chaotic temperature dependence at zero temperature

Cited by in corpus (17)

• Ergodic optimization in dynamical systems
• Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature
• Sensitive dependence of Gibbs measures at low temperatures
• Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach
• Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces
• The Analyticity of a Generalized Ruelle's Operator
• Applications of Variable Discounting Dynamic Programming to Iterated Function Systems and Related Problems
• Gibbs States and Gibbsian Specifications on the space ${\mathbb{R}}^{\mathbb{N}}$
• Generic properties of invariant measures of full-shift systems over perfect separable metric spaces
• A thermodynamic formalism for continuous time Markov chains with values on the Bernoulli Space: entropy, pressure and large deviations
• Entropy, pressure, ground states and calibrated sub-actions for linear dynamics
• Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism
• On involution kernels and large deviations principles on $\beta$-shifts
• Uniqueness and statistical properties of the Gibbs state on general one-dimensional lattice systems with markovian structure
• Zero-Temperature Chaos in Bidimensional Models with Finite-Range Potentials
• Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure
• Grand-canonical Thermodynamic Formalism via IFS: volume, temperature, gas pressure and grand-canonical topological pressure