Summary
Fractal measures are characterized by means of suitable conservation laws which can be expressed either by introducing pointwise dimensions (local approach) or by evaluating global dynamical invariants like the dimension functionD(q). The two points of view are formally equivalent to statistical mechanics and thermodynamics, respectively. Nonlinear dissipative dynamical systems are mapped onto one-dimensional Hamiltonian spin models with the introduction of appropriate statistical ensembles, using symbolic dynamics. The various thermodynamic ensembles are shown to be related to different covering procedures for fractal measures. Nonanalytic behaviour of the dimension functionD(q) is interpreted in terms of phase transitions on the time lattice. This phenomenon, occurring for non-self-similar measures, is due to long-time correlations in the symbolic dynamics and is not restricted to nonhyperbolic systems.
Riassunto
Le misure frattali vengono caratterizzate per mezzo di leggi di conservazione che possono essere espresse introducendo dimensioni puntiformi (approccio locale) o valutando invarianti dinamici globali come la funzione dimensioneD(q). I due punti di vista sono formalmente equivalenti a meccanica statistica e termodinamica, rispettivamente. Attraverso la dinamica simbolica, è possibile rappresentare i sistemi dinamici dissipativi nonlineari per mezzo di modelli di spin Hamiltoniani in reticoli unidimensionali, con l'introduzione di insiemi statistici appropriati. Si mostra che i vari insiemi termodinamici sono legati all'adozione di diversi ricoprimenti delle misure frattali. Il comportamento non analitico della funzione dimensioneD(q) viene interpretato in termini di transizioni di fase sul reticolo temporale. Questo fenomeno, che appare nelle misure non «auto-similari», è dovuto a correlazioni a tempi lunghi nella dinamica simbolica e non è ristretto ai sistemi non iperbolici.
Резюме
Фрактальные меры характеризуются с помощью соответствующих законов сохранения, которые могут быть выражены путем введения точечных размерностей (локальный подход) или путем вычисления глобальных динамических инвариантов, подобных функции размерностиD(q). Эти две точки зрения являются формально эквивалентными соответственно статистической механике и термодинамике. Нелинейные диссипативные динамические системы отображаются в одномерные гамильтоновы спиновые модели с введением соответствующих статистических ансамблей, используя символическую динамику. Показывается, что различные термодинамические ансамбли связаны с различными процедурами для фрактальных мер. Неаналитическое поведение функции размерностиD(q) интерпретируется в терминах фазовых переходов на временной решетке. Это явление, возникающее для несамоподобных мер, обусловлено длинно-временными корреляциями в символической динамике и не ограничено негиперболическими системами.
Similar content being viewed by others
References
J. P. Eckmann andD. Ruelle:Rev. Mod. Phys.,57, 617 (1985).
J. P. Eckmann:Rev. Mod. Phys.,53, 643 (1981).
M. J. Feigenbaum:J. Stat. Phys.,19, 25 (1978).
D. Ruelle andF. Takens:Commun. Math. Phys.,20, 167 (1971).
B. B. Mandelbrot:The Fractal Geometry of Nature (Freeman, San Francisco, Cal., 1982).
P. Billingsley:Ergodic Theory and Information (J. Wiley & Sons, New York, N. Y., 1965).
P. Walter:Ergodic Theory, Introductory Lectures, inLecture Notes in Mathematics, Vol.458 (Springer, Berlin, 1975).
V. I. Oseledec:Moscow Math. Soc.,19, 197 (1968).
Ya. B. Pesin:Russ. Math. Surv.,32, 55 (1977).
P. Grassberger, R. Badii andA. Politi:J. Stat. Phys.,51, 135 (1988).
R. Badii:Conservation Laws and Thermodynamic Formalism for Dissipative Dynamical Systems, to appear inRiv. Nuovo Cimento.
Ya. G. Sinai:Russ. Math. Surv.,27, 21 (1972);R. Bowen:Lecture Notes in Mathematics, Vol.470 (Springer, Berlin, 1975);D. Ruelle:Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, Vol.5 (Addison-Wesley, Reading, Mass., 1978).
O. E. Lanford: inStatistical Mechanics (CIME Lectures, 1976).
P. Cvitanovič: inProceedings of the Workshop in Condensed Matter, Atomic and Molecular Physics, Trieste, Italy, 1986.
D. Katzen andI. Procaccia:Phys. Rev. Lett.,58, 1169 (1987).
R. Badii andA. Politi: inDimensions and Entropies in Chaotic Systems, edited byG. Mayer-Kress (Springer, Berlin, 1986).
R. Badii andA. Politi:Phys. Scr.,35, 243 (1987).
J. D. Farmer, E. Ott andJ. A. Yorke:Physica D,7, 153 (1983).
V. M. Alekseev andM. V. Yakobson:Phys. Rep.,75, 290 (1981).
P. Grassberger andH. Kantz:Phys. Lett. A,113, 235 (1985).
A. Cohen andI. Procaccia:Phys. Rev. A,31, 1872 (1985).
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia andB. Shraiman:Phys. Rev. A,33, 1141 (1986).
A. Renyi:Probability Theory (North-Holland, Amsterdam, 1970).
P. Grassberger:Phys. Lett. A,97, 227 (1983).
H. G. Hentschel andI. Procaccia:Physica D,8, 435 (1983).
R. Badii andA. Politi:Phys. Rev. Lett.,52, 1661 (1984);J. Stat. Phys.,40, 725 (1985).
G. Broggi:J. Opt. Soc. Am. B,5, 1020 (1988).
R. Badii andA. Politi:Strange Attractors: Estimating the Complexity of Chaotic Signals, Nato-ASI onInstabilities and Chaos in Quantum Optics (Reidel, Dordrecht, 1988), to appear.
R. Badii andG. Broggi:Measurement of the dimension spectrum f(α): fixed-mass approach, to appear inPhys. Lett. A (1988).
A. Münster:Statistical Thermodynamics (Springer, Berlin, 1969).
G. Parisi, appendix inU. Frisch:Fully Developed Turbulence and Intermittency, inProceedings of the International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, edited byM. Ghil (North-Holland, Amsterdam, 1984);U. Frisch:Phys. Scr. T,9, 137 (1985).
P. Collet, J. Lebowitz andA. Porzio:J. Stat. Phys.,47, 609 (1987);T. Bohr andD. Rand:Physica D,25, 387 (1987);D. Rand:The Singularity Spectrum for Hyperbolic Cantor Sets and Attractors (University of Arizona, 1986), preprint.
A. S. Wightman: Introduction toConvexity in the Theory of Lattices Gases, edited byR. B. Israel (Princeton University Press, Princeton, N. J., 1979).
M. J. Feigenbaum:J. Stat. Phys.,46, 919 and 925 (1987).
M. H. Jensen, L. P. Kadanoff andI. Procaccia:Phys. Rev. A,36, 1409 (1987).
R. Badii, K. Heinzelmann, P. F. Meier andA. Politi:Phys. Rev. A,37, 1323 (1988).
T. Tél:Phys. Lett. A,119, 65 (1986).
P. Szépfalusy andT. Tél:Phys. Rev. A,34, 2520 (1986).
M. J. Feigenbaum, M. H. Jensen andI. Procaccia:Phys. Rev. Lett.,57, 1503 (1986).
M. J. Feigenbaum:Los Alamos Sci.,1, 4 (1980).
E. H. Lieb andD. C. Mattis:Mathematical Physics in One Dimension (Academic Press, New York, N. Y., 1966).
M. Hénon:Commun. Math. Phys.,50, 69 (1976).
J. Guckenheimer andP. Holmes:Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, New York, N. Y., 1986).
J. L. Kaplan andJ. A. Yorke:Lecture Notes in Mathematics,13, 730 (1979).
R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi andM. A. Rubio:Phys. Rev. Lett.,60, 979 (1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Badii, R. Generalized thermodynamic ensembles for fractal measures. Il Nuovo Cimento D 10, 819–840 (1988). https://doi.org/10.1007/BF02450142
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02450142