Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T00:24:41.340Z Has data issue: false hasContentIssue false
  • English
  • Français

Entropy production at weak Gibbs measures and a generalized variational principle

Published online by Cambridge University Press:  01 August 2009

MICHIKO YURI
MICHIKO YURI*
Affiliation:
Department of Mathematics, Graduate School of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider piecewise invertible systems exhibiting intermittency and establish a generalized variational principle adapted to a non-stationary process in the following sense; the supremum is attained by non-singular (not necessarily invariant) probability measures and if the system exhibits hyperbolicity, then it reduces to the usual variational principle for the pressure. Our method relies on Ruelle’s program in the study of non-equilibrium statistical mechanics to analyze dissipative phenomena. We show non-positivity of entropy production at weak Gibbs measures and clarify when it indeed vanishes. We also discuss a generalized variational principle in the context of σ-finite invariant measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1327 - 1347
Copyright
Copyright © Cambridge University Press 2009

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

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Gallavotti, G. and Ruelle, D.. SRB states and nonequilibrium statistical mechanics close to equilibrium. Commun. Math. Phys. 190 (1997), 279–285.CrossRefGoogle Scholar
[3]Maes, C., Reidig, F., Takens, F., van Moffaert, A. and Verbitski, E.. Intermittency and weak Gibbs states. Nonlinearity 13 (2000), 1681–1689.CrossRefGoogle Scholar
[4]Pesin, Y. and Santi, S.. Themodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J. 5(3) (2005), 669–678.CrossRefGoogle Scholar
[5]Pesin, Y. and Santi, S.. Equilibrium measures for maps with inducing schemes. J. Modern. Dynam. 2(3) (2008), 1–31.Google Scholar
[6]Pesin, Y. B. and Zhang, Ke. Thermodynamics associated with inducing schemes and liftability of measures. Proc. Fields Inst. 51 (2007), 289–305.Google Scholar
[7]Ruelle, D.. Entropy production in nonequilibrium statistical mechanics. Commun. Math. Phys. 189 (1997), 365–371.CrossRefGoogle Scholar
[8]Ruelle, D.. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95 (1999), 393–468.CrossRefGoogle Scholar
[9]Schweiger, F.. Ergodic Theory and Fibred Systems and Metric Number Theory. Oxford University Press, Oxford, 1995.Google Scholar
[10]Thaler, M.. Transformations on [0, 1] with infinite invariant measures. Israel J. Math. 46 (1983), 67–97.CrossRefGoogle Scholar
[11]Yuri, M.. Multi-dimensional maps with infinite invariant measures and countable state sofic shifts. Indag. Math. (N.S.) 6 (1995), 355–383.CrossRefGoogle Scholar
[12]Yuri, M.. Zeta functions for certain nonhyperbolic systems and topological Markov approximations. Ergod. Th. & Dynam. Sys. 18 (1998), 1589–1612.CrossRefGoogle Scholar
[13]Yuri, M.. Thermodynamic formalism for certain nonhyperbolic maps. Ergod. Th. & Dynam. Sys. 19 (1999), 1365–1378.CrossRefGoogle Scholar
[14]Yuri, M.. Weak Gibbs measures for certain nonhyperbolic systems. Ergod. Th. & Dynam. Sys. 20 (2000), 1495–1518.CrossRefGoogle Scholar
[15]Yuri, M.. On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points. Nonlinearity 15 (2002), 429–445.CrossRefGoogle Scholar
[16]Yuri, M.. Multifractal analysis of weak Gibbs measures for intermittent systems. Commun. Math. Phys. 230 (2002), 365–388.CrossRefGoogle Scholar
[17]Yuri, M.. Thermodynamic Formalism for countable to one Markov systems. Trans. Amer. Math. 355 (2003), 2949–2971.CrossRefGoogle Scholar
[18]Yuri, M.. Weak Gibbs measures for Intermittent systems and weakly Gibbsian states in Statistical mechanics. Commun. Math. Phys. 241 (2003), 453–466.CrossRefGoogle Scholar