Access by Xinjiang University

Thermodynamic phase transitions for Pomeau-Manneville maps

Centro de Matemática, Computação e Cognição, UFABC, 09210-170, Santo André, SP, Brazil

^*roberto.venegeroles@ufabc.edu.br

Phys. Rev. E 86, 021114 – Published 14 August, 2012

DOI: https://doi.org/10.1103/PhysRevE.86.021114

Abstract

We study phase transitions in the thermodynamic description of Pomeau-Manneville intermittent maps from the point of view of infinite ergodic theory, which deals with diverging measure dynamical systems. For such systems, we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase. In particular, topological pressure and Rényi entropy are calculated exactly for such systems. Finally, we show the connection of the distributional limit theorem with non-Gaussian fluctuations of the algorithmic complexity proposed by Gaspard and Wang [Proc. Natl. Acad. Sci. USA 85, 4591 (1988)].

Article Text

References (33)

L. E. Boltzmann, S. G. BrushVorlesungen uber Gastheorie, 2 vols. (J. A. Barth Verlag, Leipzig, 1895-1898), English translation by [Lectures on Gas Theory (Dover, New York, 1995)].
Ya. G. Sinai, Russian Math. Surveys 27, 21 (1972).
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470 (Springer, New York, 1975).
D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978; second edition by Cambridge University Press, Cambridge, England, 2004).
C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems: An Introduction (Cambridge University Press, Cambridge, England, 1995).
M. J. Feigenbaum, M. H. Jensen, and I. Procaccia, Phys. Rev. Lett. 57, 1503 (1986).
D. Katzen and I. Procaccia, Phys. Rev. Lett. 58, 1169 (1987).
T. Bohr and M. H. Jensen, Phys. Rev. A 36, 4904 (1987).
R. Badii and A. Politi, Phys. Rev. A 35, 1288 (1987).
P. Grassberger, R. Badii, and A. Politi, J. Stat. Phys. 51, 135 (1988).
P. Szépfalusy, T. Tél, A. Csordás, and Z. Kovács, Phys. Rev. A 36, 3525 (1987).
X.-J. Wang, Phys. Rev. A 39, 3214 (1989); 40, 6647 (1989).
M. J. Feigenbaum, J. Stat. Phys. 52, 527 (1989).
N. Mori, T. Kobayashi, H. Hata, T. Morita, T. Horita, and H. Mori, Prog. Theor. Phys. 81, 69 (1989).
S. Sato and K. Honda, Phys. Rev. A 42, 3233 (1990).
T. Prellberg and J. Slawny, J. Stat. Phys. 66, 503 (1992).
Y. Pomeau and P. Manneville, Commun. Math. Phys. 74, 189 (1980).
M. Thaler, Studia Math. 143, 103 (2001).
A. Saa and R. Venegeroles, J. Stat. Mech.: Theory Exp. (2012) P03010.
P. Gaspard and X.-J. Wang, Proc. Natl. Acad. Sci. USA 85, 4591 (1988).
W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1971), Vol. II.
J. R. Rigby and A. Porporato, Adv. Water Resour. 33, 923 (2010); C. Nicolis, W. Ebeling, and C. Baraldi, Tellus, Ser. A 49, 108 (1997).
W. Li and K. Kaneko, Europhys. Lett. 17, 655 (1992); Nature (London) 360, 635 (1992).
W. Ebeling and G. Nicolis, Europhys. Lett. 14, 191 (1991).
J. Aaronson, An Introduction to Infinite Ergodic Theory (American Mathematical Society, Providence, 1997).
C. J. A. Pires, A. Saa, and R. Venegeroles, Phys. Rev. E 84, 066210 (2011).
M. E. Fisher and B. U. Felderhof, Ann. Phys. (NY) 58, 176 (1970); 58, 217 (1970); 58, 268 (1970); 58, 281 (1970).
J. P. Eckmann and I. Procaccia, Phys. Rev. A 34, 659 (1986).
A. Saa and R. Venegeroles, Phys. Rev. E 84, 026702 (2011).
K. A. Penson and K. Gorska, Phys. Rev. Lett. 105, 210604 (2010).
R. Zweimüller, Ergod. Th. Dynam. Sys. 20, 1519 (2000).
M. G. Mittag-Leffler, Acta Math. 29, 101 (1905).
From the series definition of the Mittag-Leffler function, it is easy to see that $E_{γ} (- u) = 2 E_{2 γ} (u^{2}) - E_{γ} (u)$ , thus $E_{γ} (u) \sim 0$ for $u \to - \infty$ .

Physical Review E

Export citation

Export citation

Thermodynamic phase transitions for Pomeau-Manneville maps

Abstract

Article Text

References (33)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation