Entropy and variational principle for one-dimensional lattice systems with a generala prioriprobability: positive and zero temperature
- 3 July 2014
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 35 (6), 1925-1961
- https://doi.org/10.1017/etds.2014.15
Abstract
We generalize several results of the classical theory of thermodynamic formalism by considering a compact metric space $M$ as the state space. We analyze the shift acting on $M^{\mathbb{N}}$ and consider a generala prioriprobability for defining the transfer (Ruelle) operator. We study potentials $A$ which can depend on the infinite set of coordinates in $M^{\mathbb{N}}$ . We define entropy and by its very nature it is always a non-positive number. The concepts of entropy and transfer operator are linked. If $M$ is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case $M=S^{1}$ , and thea priorimeasure is Lebesgue $dx$ , the infinite product of $dx$ on $(S^{1})^{\mathbb{N}}$ will have zero entropy. We analyze the Pressure problem for a Hölder potential $A$ and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the thermodynamic formalism for the Bernoulli space with countable infinite symbols. Moreover, the so-called $XY$ model also fits under our setting. In this last case $M$ is the unitary circle $S^{1}$ . We explore the differentiable structure of $(S^{1})^{\mathbb{N}}$ by considering a certain class of smooth potentials and we show some properties of the corresponding main eigenfunctions.
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This publication has 43 references indexed in Scilit:
- On the selection of subaction and measure for a subclass of potentials defined by P. WaltersErgodic Theory and Dynamical Systems, 2012
- Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentialsErgodic Theory and Dynamical Systems, 2010
- Negative entropy, zero temperature and Markov chains on the intervalBulletin of the Brazilian Mathematical Society, New Series, 2009
- On the Aubry–Mather theory for symbolic dynamicsErgodic Theory and Dynamical Systems, 2008
- Entropy penalization methods for Hamilton–Jacobi equationsAdvances in Mathematics, 2007
- Ergodic Optimization for Renewal Type ShiftsMonatshefte für Mathematik, 2006
- A dynamical proof for the convergence of Gibbs measures at temperature zeroNonlinearity, 2005
- Sub-actions for Anosov flowsErgodic Theory and Dynamical Systems, 1999
- Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theoryJournal of Statistical Physics, 1993
- Infinite-Spin Ising Model in One DimensionJournal of Mathematical Physics, 1968