Advertisement

Communications in Mathematical Physics

, Volume 267, Issue 3, pp 631–667 | Cite as

Continuous Phase Transitions for Dynamical Systems

  • Omri Sarig
Article

Abstract

We study the asymptotic expansion of the topological pressure of one–parameter families of potentials at a point of non-analyticity. The singularity is related qualitatively and quantitatively to non–Gaussian limit laws and to slow decay of correlations with respect to the equilibrium measure.

Keywords

Critical Exponent Gibbs Measure Equilibrium Measure Thermodynamic Formalism Continuous Phase Transition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ADU.
    Aaronson J., Denker M., Urbański M. (1993). Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. AMS 337:495–548CrossRefzbMATHGoogle Scholar
  2. AD.
    Aaronson J., Denker M. (2001). Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1(2):193–237CrossRefMathSciNetzbMATHGoogle Scholar
  3. BG.
    Bálint P., Gouëzel S. (2006). Limit theorems in the stadium billiard. Commun. Math. Phys. 263, 2, 461–512Google Scholar
  4. BGT.
    Bingham N.H., Goldie C.M., Teugels J.L. (1987). Regular variation. Encyclopedia of Math. and its Appl. 27, Cambridge Univ. Press, CambridgezbMATHGoogle Scholar
  5. BDFN.
    Binney J.J., Dowrick N.J., Fisher A.J., Newman M.E.J. (1992). The theory of critical phenomena, an introduction to the renormalization group. Oxford Science Publications, Oxford University Press, OxfordzbMATHGoogle Scholar
  6. Bo.
    Bowen R. (1975). Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New YorkzbMATHGoogle Scholar
  7. BS.
    Buzzi J., Sarig O. (2003). Uniqueness of equilibrium measures for countable Markov shifts and multi-dimensional piecewise expanding maps. Erg. Th. Dynam. Sys. 23:1383–1400CrossRefMathSciNetzbMATHGoogle Scholar
  8. Ea.
    Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. (Russian) Teor. Verojatnost. i Primenen. 21(3), 653–660 (1976) Engl. Transl. in Theor. Probab. Appl. 21(3), 637–642 (1976)Google Scholar
  9. El.
    Ellis R.S. (1985). Entropy, large deviations, and statistical mechanics. Grund. Math. Wissenschaften 271, Springer Verlag, Berlin Heidelberg-NewyorkzbMATHGoogle Scholar
  10. F.
    Feller W. (1971). An introduction to probability theory and its applications Volume II, Second edition. John Wiley & Sons, NewyorkGoogle Scholar
  11. FF.
    Fisher M.E., Felderhof B.U. (1970). Phase transition in one–dimensional cluster–interaction fluids: IA. Thermodynamics, IB. Critical behavior. II. Simple logarithmic model. Ann. Phy. 58:177–280ADSGoogle Scholar
  12. GK.
    Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Translated and annotated by K.L. Chung, with an Appendix by J.L. Doob. Readings MA: Addison–Wesley Publishing Company, 1954Google Scholar
  13. Gou1.
    Gouëzel S. (2004). Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139:29–65CrossRefMathSciNetzbMATHGoogle Scholar
  14. Gou2.
    Gouëzel S. (2004). Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128(1):82–122CrossRefMathSciNetzbMATHGoogle Scholar
  15. Gu1.
    Gurevič B.M. (1969). Topological entropy of a countable Markov chain. Dokl. Akad. Nauk SSSR 187:715–718MathSciNetGoogle Scholar
  16. Gu2.
    Gurevich B.M. (1984). A variational characterization of one-dimensional countable state Gibbs Random field. Z. ahrscheinlichkeitstheorie verw. Gebiete 68:205–242CrossRefMathSciNetzbMATHGoogle Scholar
  17. H1.
    Haydn N., Isola S. (2001). Parabolic rational maps. J. London Math. Soc. 63(2):673–689CrossRefMathSciNetzbMATHGoogle Scholar
  18. Hi.
    Hilfer R. (1993). Classification theory for anequilibrium phase transitions. Phys. Rev. E 48(4):2466–2475CrossRefADSMathSciNetGoogle Scholar
  19. Ho.
    Hofbauer F. (1977). Examples for the non–uniqueness of the equilibrium states. Trans. AMS 228:223–241CrossRefMathSciNetzbMATHGoogle Scholar
  20. Ka.
    Kato T. Perturbation theory for linear operators Reprint of the 1980 edition. Classics in Mathematics.Berlin: Springer-Verlag, (1995).Google Scholar
  21. Ke.
    Keane M. (1972). Strongly mixing g-measures. Invent. Math. 16:309–324CrossRefADSMathSciNetzbMATHGoogle Scholar
  22. Lo.
    Lopes A.O. (1993). The Zeta function, non–differentiability of pressure, and the critical exponent of transition. Adv. Math. 101(2):133–165CrossRefMathSciNetzbMATHGoogle Scholar
  23. ML.
    Martin–Löf A. (1973). Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Commun. Math. Phys. 32:75–92CrossRefADSGoogle Scholar
  24. MU1.
    Urbański M., Mauldin R.D. (2001). Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125:93–130CrossRefMathSciNetzbMATHGoogle Scholar
  25. MU2.
    Mauldin, R.D., Urbański, M.: Graph directed Markov systems; Geometry and dynamics of limit sets. Cambridge Tracts in Mathematics, 148, Cambridge: Cambridge University Press, Cambridge, 2003.Google Scholar
  26. MT.
    Melbourne I., Török A. (2004). Statistical limit theorems for suspension flows. Israel J. Math. 194:191–210CrossRefGoogle Scholar
  27. N.
    Nagaev S.V. (1957). Some limit theorems for stationary Markov chains. (Russian) Teor. Veroyatnost. i Primenen. 2:389–416MathSciNetGoogle Scholar
  28. PS.
    Prellberg T., Slawny J. (1992). Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2):503–514CrossRefADSMathSciNetzbMATHGoogle Scholar
  29. Ru.
    Ruelle D. (2004). Thermodynamic Formalism, The mathematical structures of equilibrium statistical mechanics. 2nd Ed. Cambridge Mathematical Library. Cambridge University Press, CambridgezbMATHGoogle Scholar
  30. S1.
    Sarig O.M. (1999). Thermodynamic Formalism for Countable Markov shifts. Erg. Th. Dyn. Sys. 19:1565–1593CrossRefMathSciNetzbMATHGoogle Scholar
  31. S2.
    Sarig O. (2001). Phase Transitions for Countable Markov Shifts. Commun. Math. Phys. 217:555–577CrossRefADSMathSciNetzbMATHGoogle Scholar
  32. S3.
    Sarig O. (2003). Characterization of existence of Gibbs measures for Countable Markov shifts. Proc. of AMS. 131(6):1751–1758CrossRefMathSciNetzbMATHGoogle Scholar
  33. S4.
    Sarig O. (2001). Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121:285–311CrossRefMathSciNetzbMATHGoogle Scholar
  34. S5.
    Sarig O. (2002). Subexponential decay of correlations. Invent. Math. 150:629–653CrossRefMathSciNetzbMATHGoogle Scholar
  35. S6.
    Sarig, O.: Thermodynamic formalism for countable Markov shifts. Tel-Aviv University Thesis (2000).Google Scholar
  36. S7.
    Sarig O. (2000). On an example with a non-analytic topological pressure. C. R. Acad. Sci. Paris Sér. I Math. 330(4):311–315MathSciNetzbMATHGoogle Scholar
  37. St.
    Stanley H.E. (1971). Introduction to phase transitions and critical phenomena. Oxford University Press, OxfordGoogle Scholar
  38. W.
    Walters P. (1975). Ruelle’s operator theorem and g-measures. Trans. Amer. Math. Soc. 214:375–387CrossRefMathSciNetzbMATHGoogle Scholar
  39. Wa1.
    Wang, X.-J.: Abnormal fluctuations and thermodynamic phase transition in dynamical systems. Phys. Review A 39(6), 3214–3217Google Scholar
  40. Wa2.
    Wang X.-J. (1989). Statistical physics of temporal intermittency. Phys. Review A 40(11):6647–6661CrossRefADSGoogle Scholar
  41. Y1.
    Yuri M. (2003). Thermodynamic formalism for countable to one Markov systems. Trans. Amer. Math. Soc. 355(7):2949–2971CrossRefMathSciNetzbMATHGoogle Scholar
  42. Y2.
    Yuri M. (2005). Phase transition, non-Gibbsianness and subexponential instability, Ergodic Thy Dynam. Syst. 25:1325–1342MathSciNetzbMATHGoogle Scholar
  43. Z.
    Zolotarev V.M. (1986). One–dimensional stable distributions. Transl. Math. Monog. 65, Amer. Math. Sec., Providence, RIzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematics DepartmentPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations