Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-02T00:58:38.458Z Has data issue: false hasContentIssue false
  • English
  • Français

Zero-temperature phase diagram for double-well type potentials in the summable variation class

Published online by Cambridge University Press:  19 September 2016

RODRIGO BISSACOT ,
EDUARDO GARIBALDI  and
PHILIPPE THIEULLEN
RODRIGO BISSACOT
Affiliation:
Department of Applied Mathematics, University of Sao Paulo, 05508-090 Sao Paulo, Brazil email rodrigo.bissacot@gmail.com
EDUARDO GARIBALDI
Affiliation:
Department of Mathematics, University of Campinas, 13083-859 Campinas, Brazil email garibaldi@ime.unicamp.br
PHILIPPE THIEULLEN
Affiliation:
Institut de Mathématiques, Université de Bordeaux, CNRS, UMR 5251, F-33405 Talence, France email Philippe.Thieullen@math.u-bordeaux1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the zero-temperature limit of the Gibbs measures of a class of long-range potentials on a full shift of two symbols $\{0,1\}$ . These potentials were introduced by Walters as a natural space for the transfer operator. In our case, they are constant on a countable infinity of cylinders and are Lipschitz continuous or, more generally, of summable variation. We assume that there exist exactly two ground states: the fixed points $0^{\infty }$ and $1^{\infty }$ . We fully characterize, in terms of the Peierls barrier between the two ground states, the zero-temperature phase diagram of such potentials, that is, the regions of convergence or divergence of the Gibbs measures as the temperature goes to zero.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 863 - 885
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atkinson, G.. Recurrence of co-cycles and random walks. J. Lond. Math. Soc. 2 (1976), 486–488.Google Scholar
Aubrun, N. and Sablik, M.. Simulation of effective subshifts by two-dimensional subshifts of finite type. Acta Appl. Math. 126 (2013), 35–63.Google Scholar
Baladi, V.. Positive Transfer Operator and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16) . World Scientific, Singapore, 2000.Google Scholar
Baraviera, A., Leplaideur, R. and Lopes, A. O.. Selection of ground states in the zero temperature limit for a one-parameter family of potentials. SIAM J. Appl. Dyn. Sys. 11 (2012), 243–260.CrossRefGoogle Scholar
Baraviera, A., Lopes, A. O. and Mengue, J. K.. On the selection of subaction and measure for a subclass of potentials defined by P. Walters. Ergod. Th. & Dynam. Sys. 33 (2013), 1338–1362.CrossRefGoogle Scholar
Brémont, J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419–426.Google Scholar
Chazottes, J. R., Gambaudo, J.-M. and Ulgade, E.. Zero-temperature limit of one dimensional Gibbs states via renormalization: the case of locally constant potentials. Ergod. Th. & Dynam. Sys. 31 (2011), 1109–1161.Google Scholar
Chazottes, J. R. and Hochman, M.. On the zero-temperature limit of Gibbs states. Commun. Math. Phys. 297 (2010), 265–281.Google Scholar
Coronel, D. and Rivera-Letelier, J.. Sensitive dependence of Gibbs measures. J. Stat. Phys. 160 (2015), 1658–1683.Google Scholar
van Enter, A. C. D. and Ruszel, W. M.. Chaotic temperature dependence at zero temperature. J. Stat. Phys. 127 (2007), 567–573.CrossRefGoogle Scholar
Freire, R. and Vargas, V.. Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts. Preprint, 2015, arXiv:1511.01527.Google Scholar
Garibaldi, E., Lopes, A. O. and Thieullen, Ph.. On calibrated and separating sub-actions. Bull. Braz. Math. Soc. 40 (2009), 577–602.CrossRefGoogle Scholar
Garibaldi, E. and Thieullen, Ph.. Description of some ground sates by Puiseux techniques. J. Stat. Phys. 146 (2012), 125–180.Google Scholar
Hochman, M.. On the dynamics and recursive properties of multidimensional symbolic system. Invent. Math. 176 (2009), 131–167.Google Scholar
Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Students Texts, 42) . Cambridge University Press, Cambridge, 1998.Google Scholar
Kempton, T.. Zero temperature limits of Gibbs equilibrium states for countable Markov shifts. J. Stat. Phys. 143 (2011), 795–806.CrossRefGoogle Scholar
Leplaideur, R.. A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18 (2005), 2847–2880.Google Scholar
Leplaideur, R.. Flatness is a criterion for selection of maximizing measures. J. Stat. Phys. 147 (2012), 728–757.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
Ruelle, D.. Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn. Cambridge University Press, Cambridge, 2004.Google Scholar
Walters, P.. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys. 27 (2007), 1323–1348.Google Scholar