Abstract
These lecture notes illustrate some features of deterministic transport in chaotic systems. The subject has witnessed an impressive amount of work in the last thirty years, and our review is not meant to be exhaustive, but rather focus on some unifying techniques by which the problem can be tackled, pointing out difficulties and open problems.
We start by dealing with the case of hyperbolic systems where typically normal diffusion is observed (even though actual calculation of transport coefficients may be exceedingly difficult), while the second part of the notes deals with weakly chaotic systems, where long trappings near regular phase-space regions may induce anomalies in diffusive properties. Examples of analytic calculations are given in the framework of cycle expansions, a general technique for getting chaotic averages.
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References
Klages, R., Dorfman, J.R. (1995): Simple maps with fractal diffusion coefficient. Phys. Rev. Lett., 74, 387–390
Artuso R. (1991): Diffusive dynamics and periodic orbits of dynamical systems. Phys. Lett., A 160, 528–530
Cvitanović, P., Gaspard, P., Schreiber T. (1992): Investigation of the Lorentz Gas in terms of periodic orbits. CHAOS, 2, 85–90
Cvitanović, P., Eckmann, J.-P., Gaspard P. (1995): Transport properties of the Lorentz gas in terms of periodic orbits. Chaos, Solitons and Fractals, 6, 113–120
Dana, I. (1989): Hamiltonian transport on unstable periodic orbits. Physica, D 39, 205–230
Vance, W.N. (1992): Unstable periodic orbits and transport properties of nonequilibrium steady states. Phys. Rev. Lett., 96, 1356–1359
Artuso, R., Aurell, E., Cvitanović, P. (1990): Recycling of strange sets I: Cycle expansions. Nonlinearity 3, 325–360
Artuso, R., Aurell, E., Cvitanović, P. (1990): Recycling of strange sets II: Applications. Nonlinearity 3, 361–386
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G. (2001): Classical and Quantum Chaos. http://www.nbi.dk/ChaosBook/, Niels Bohr Institute, Copenhagen
Baladi, V. (1995): Dynamical zeta functions. In: Branner, B., Hjorth (eds) Proceedings of the NATO ASI Real and Complex Dynamical Systems. Kluwer Academic Publishers Dordrecht
Devaney, R.L. (1987): An Introduction to Chaotic Dynamical Systems. Addison-Wesley Reading MA
Hansen, K.T. (1994): Symbolic Dynamics in Chaotic Systems. Ph.D. thesis, University of Oslo http://www.nbi.dk/CATS/papers/khansen/thesis/thesis.html
Artuso, R., Strepparava, R. (1997): Recycling diffusion in sawtooth and cat maps. Phys. Lett. A236 469–475
Arnol'd V.I., Avez, A. (1967): Problèmes Ergodiques de la Mécanique Classique. Gauthier-Villars Paris
Rugh, H.H. (1992): The Correlation Spectrum for Hyperbolic Analytic Maps. Nonlinearity 5, 1237–1263
Percival, I., Vivaldi, F. (1987): A linear code for the sawtooth and cat maps. Physica 27D, 373–386
Bird, N., Vivaldi, F. (1988): Periodic orbits of the sawtooth maps. Physica D30, 164–176
Coxeter, H.S.M. (1948): Regular Polytopes. Methuen London
Lichtenberg, A.J., Lieberman, M.A. (1982): Regular and stochastic motion. Springer New York
Cary, J.R., Meiss, J.D. (1981): Rigorously diffusive deterministic map. Phys. Rev. A24, 2664–2668
Pomeau, Y., Manneville, P.(1980): Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197
Gaspard, P., Wang, X.-J. (1988): Sporadicity: between periodic and chaotic dynamical behavior. Proc. Natl. Acad. Sci. U.S.A. 85 4591–4595
Feller, W. (1966): An introduction to probability theory and applications, Vol. II. Wiley New York
Geisel, T., Thomae, S. (1984): Anomalous diffusion in intermittent chaotic systems. Phys. Rev. Lett. 52 1936–1939
Artuso, R., Casati G., Lombardi, R. (1993): Periodic orbit theory of anomalous diffusion. Phys. Rev. Lett. 71 62–64
Dahlqvist, P. (1995): Approximate zeta functions for the Sinai billiard and related systems. Nonlinearity 8, 11–28
Baladi, V., Eckmann, J.-P., Ruelle, D. (1989): Resonances for intermittent systems. Nonlinearity 2, 119–135
Dahlqvist, P., Artuso, R. (1996): On the decay of correlations in the Sinai billiard with infinite horizon. Phys. Lett. A 219, 212–216
Procaccia, I., Schuster, H. (1983): Functional renormalization group theory of universal 1/f noise in dynamical systems. Phys. Rev. A 28, 1210–1212
Dahlqvist, P. (1996): Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon. J. Stat. Phys. 84 773–795
Karney, C.F.F. (1983): Long-time correlations in the stochastic regime. Physica D 8, 360–380
Chirikov, B.V., Shepelyansky, D.L. (1984): Correlation properties of dynamical chaos in Hamiltonian systems. Physica D 13, 395–400
Artuso, R. (1999): Correlation decay and return time statistics. Physica 131 68–77
Young, L.-S. (1999): Recurrence times and rates of mixing. Israel J. Math. 110 153–188
Bunimovich, L.A. (1985): Decay of correlations in dynamical systems with chaotic behavior. Sov. Phys. JETP 62 842–852
Bleher, P.M. (1992): Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. J. Stat. Phys. 66 315–373
Artuso, R., Casati, G., Guarneri, I. (1996): Numerical experiments on billiards. J. Stat. Phys. 83 145–166
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Artuso, R. (2003). From Normal to Anomalous Deterministic Diffusion. In: Esposti, M.D., Graffi, S. (eds) The Mathematical Aspects of Quantum Maps. Lecture Notes in Physics, vol 618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-37045-5_5
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