Prog. Theor. Phys. Vol. 71 No. 4 (1984) pp. 851-854

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Letters

Towards the Statistical Mechanics of Chaos

Department of Mathematics, College of Arts and Sciences, University of Tokyo, Tokyo 153
Department of Physics and Materials Reseach Laboratory, University of Illinois at Urbana-Champaign, Urbara, Illinois 61801

(Received December 13, 1983)

Abstract:

A statistical mechanical formalism for chaos is proposed. In this formalism, internal energy corresponds to the Lyapounov characteristic number and entropy to the Kolmogorov entropy. The so-called window phenomenon is related to the first order phase transition, and the Feigenbaum critical phenomena are exactly the critical phenomena within this formalism. The inverse temperature is related to the dimension of the invariant set.

URL : http://ptp.ipap.jp/link?PTP/71/851/
DOI : 10.1143/PTP.71.851

References:

See, for example, Y. Kuramoto, Physica 106A (1981), 128.
H. Swinney and J. B. Gollub ed. Hydrodynamic Instabilities and the Transition to Turbulence (Springer, NY, 1981).
See, for example, P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems (Birkhauser, Boston, 1980).
J. P. Eckmann, Rev. Mod. Phys. 53 (1981), 643[APS].
Y. Oono, Prog. Theor. Phys. 59 (1978), 1029[PTP].
Y. Oono and M. Osikawa, Prog. Theor. Phys. 64 (1980), 54[PTP].
M. Osikawa and Y. Oono, Publ. RIMS 17 (1981), 165.
Y. Takahashi, Sci. Papers College Gen. Ed. Univ. Tokyo 30 (1980), 11.
T. Li and J. A. Yorke, Am. Math. Monthly 82 (1975), 985.
e.g., S. Grossmann and S. Thomae, Z. Naturforsch. 32a (1977), 1353.
H. Fujisaka and T. Yamada, Z. Naturforsch. 33a (1978), 1455.
S. Thomae and S. Grossmann, J. Stat. Phys. 26 (1981), 485.
H. Mori, B. -C. So and T. Ose, Prog. Theor. Phys. 66 (1981), 1266[PTP].
T. Kai and K. Tomita, Prog. Theor. Phys. 64 (1980), 1532[PTP].
Y. Oono and Y. Takahashi, Prog. Theor. Phys. 63 (1980), 1804[PTP].
S. -J. Chang nad J. Wright, Phys. Rev. A23 (1981), 1419[APS].
e.g., P. Collet, J. Crutchfield and J. P. Eckmann, Commun. Math. Phys. 88 (1983), 257[CrossRef].
Y. Takahashi, Sci. Papers College Gen. Ed. Univ. Tokyo 31 (1981), 61.
Y. Takahashi, 4th Japan-USSR Symposium on Probability and Math. Statistics, Lecture Notes in Mathematics, to be published.
See, e.g., D. Ruelle, Thermodynamic Formalism (Addison-Wesley, 1978).
Y. Sinai, Russ. Math. Surveys 27 (1972), 21.
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer, 1975).
Y. Takahashi, Osaka J. Math. to be published.
a)Y. Takahashi, Lecture given at Tsuda College, April 1979.
b)Y. Oono, notes distributed in August 1981.
F. Ledrapier, Commun. Math. Phys. 81 (1981), 229[CrossRef].
Y. Oono, Prog. Theor. Phys. 60 (1978), 1944[PTP].
See also, Y. Oono, T. Kohda and H. Yamazaki, J. Phys. Soc. Jpn. 48 (1980), 738[JPSJ].
R. Bowen, Trans. AMS 184 (1973), 125.
E. I. Dinaburg, Math. USSR Izv. 5 (1971), 337.
D. Ruelle, Ergod, Theor. & Dynam, Syst. 2 (1982), 99.
M. Widom, D. Bensimon, L. P. Kadanoff and S. J. Scott, Chicago Preprint.
P. Billingsley, Ergodic Theory and Information (Wiley, NY, 1960).
R. May, Nature 261 (1976), 459[CrossRef].

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