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Relations with the Thermodynamic Formalism

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Ergodic Optimization in the Expanding Case

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Abstract

Perhaps one of the most interesting and fruitful applications of ergodic optimization theory occurs in the study of freezing phenomena in equilibrium statistical mechanics. In this concluding chapter, we provide a first glimpse of such a rich interaction among theories, by scrutinizing with basic techniques the convergence of equilibrium states to a particular maximizing probability on certain examples.

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Bibliography

  1. Aizenman, M., Lieb, E.H.: The third law of thermodynamics and the degeneracy of the ground state for lattice systems. J. Stat. Phys. 24, 279–297 (1981)

    Article  MathSciNet  Google Scholar 

  2. Akian, M., Bapat, R., Gaubert, S.: Asymptotics of the Perron eigenvalue and eigenvector using max-algebra. C. R. Acad. Sci. Ser. I 327, 927–932 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Akian, M., Bapat, R., Gaubert, S.: Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Visik-Ljusternik theorem (2006, preprint)

    Google Scholar 

  4. Baraviera, A.T., Leplaideur, R., Lopes, A.O.: Selection of measures for a potential with two maxima at the zero temperature limit. SIAM J. Appl. Dyn. Syst. 11, 243–260 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baraviera, A.T., Leplaideur, R., Lopes, A.O.: Ergodic optimization, zero temperature limits and the max-plus algebra. 29∘ Colóquio Brasileiro de Matemática. IMPA, Rio de Janeiro (2013)

    Google Scholar 

  6. Bissacot, R., Freire Júnior, R.S.: On the existence of maximizing probabilities for irreducible countable Markov shifts: a dynamical proof. Ergodic Theory Dyn. Syst. 34, 1103–1115 (2014)

    Article  MATH  Google Scholar 

  7. Bissacot, R., Garibaldi, E.: Weak KAM methods and ergodic optimal problems for countable Markov shifts. Bull. Braz. Math. Soc. New Ser. 41, 321–338 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bissacot, R., Garibaldi, E., Thieullen, P.: Zero-temperature phase diagram for double-well type potentials in the summable variation class. Ergodic Theory Dyn. Syst. (2016). doi:10.1017/etds.2016.57

    Google Scholar 

  9. Brémont, J.: Gibbs measures at temperature zero. Nonlinearity 16, 419–426(2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chazottes, J.R., Hochman, M.: On the zero-temperature limit of Gibbs states. Commun. Math. Phys. 297, 265–281 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chazottes, J.R., Gambaudo, J.M., Ugalde, E.: Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials. Ergodic Theory Dyn. Syst. 31, 1109–1161 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coronel, D., Rivera-Letelier, J.: Sensitive dependence of Gibbs measures at low temperatures. J. Stat. Phys. 160, 1658–1683 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Garibaldi, E., Thieullen, P.: Description of some ground states by Puiseux techniques. J. Stat. Phys. 146, 125–180 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gomes, J.T.A.: Formalismos gibbsianos para sistemas de spins unidimensionais. Master’s thesis, University of Campinas (2012)

    Google Scholar 

  15. Iommi, G., Yayama, Y.: Zero temperature limits of Gibbs states for almost-additive potentials. J. Stat. Phys. 155, 23–46 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jenkinson, O., Mauldin, R.D. Urbański, M.: Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type. J. Stat. Phys. 119, 765–776 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kempton, T.: Zero temperature limits of Gibbs equilibrium states for countable Markov shifts. J. Stat. Phys. 143, 795–806 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leplaideur, R.: A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18, 2847–2880 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Morris, I.D.: Entropy for zero-temperature limits of Gibbs-equilibrium states for countable-alphabet subshifts of finite type. J. Stat. Phys. 126, 315–324 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morris, I.D.: A note on approximating the maximum ergodic average via the Ruelle pressure functional (2009, preprint)

    Google Scholar 

  21. Nekhoroshev, N.N.: Asymptotics of Gibbs measures in one-dimensional lattice models. Mosc. Univ. Math. Bull. 59, 10–15 (2004)

    MathSciNet  MATH  Google Scholar 

  22. van Enter, A.C.D., Ruszel, W.M.: Chaotic temperature dependence at zero temperature. J. Stat. Phys. 127, 567–573 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Campinas - IMECC, Campinas, São Paulo, Brazil

    Eduardo Garibaldi

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  1. Eduardo Garibaldi
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Garibaldi, E. (2017). Relations with the Thermodynamic Formalism. In: Ergodic Optimization in the Expanding Case. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-66643-3_9

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