The ergodic hierarchy, randomness and Hamiltonian chaos. (English) Zbl 1223.37010
Summary: Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ergodic hierarchy (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems.
MSC:
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 37-03 | History of dynamical systems and ergodic theory |
| 01A60 | History of mathematics in the 20th century |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |
References:
| [1] | Alekseev, V. M.; Yakobson, M. V., Symbolic dynamics and hyperbolic dynamical systems, Physics Reports, 75, 287-325 (1981) |
| [2] | Argyris, J.; Faust, G.; Haase, M., An exploration of chaos (1994), Elsevier: Elsevier Amsterdam · Zbl 0805.58001 |
| [3] | Arnol’d, V. I., Proof of a theorem of A.N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspekhi Matematicheskikh Nauk, 18, 5, 113, 13-40 (1963), (Russian) · Zbl 0129.16606 |
| [4] | Arnol’d, V. I.; Avez, A., Ergodic problems of classical mechanics (1968), W. A. Benjamin: W. A. Benjamin New York and Amsterdam · Zbl 0167.22901 |
| [5] | Belot, G.; Earman, J., Chaos out of order: Quantum mechanics, the correspondence principle and chaos, Studies in the History and Philosophy of Modern Physics, 28, 147-182 (1997) · Zbl 1222.81181 |
| [6] | Boltzmann, L., Einige allgemeine Sätze über Wärmegleichgewicht, Wiener Berichte, 63, 670-711 (1871), (German) · JFM 03.0536.01 |
| [7] | Brudno, A. A., The complexity of the trajectory of a dynamical system, Russian Mathematical Surveys, 33, 197-198 (1978) · Zbl 0408.28017 |
| [8] | Chernov, N. I.; Haskell, C., Nonuniformly hyperbolic K-systems are Bernoulli, Ergodic Theory and Dynamical Systems, 16, 19-44 (1996) · Zbl 0853.58081 |
| [9] | Cornfeld, I. P.; Fomin, S. V.; Sinai, Y. G., Ergodic theory (1982), Springer: Springer Berlin and New York · Zbl 0493.28007 |
| [10] | Earman, J.; Redei, M., Why ergodic theory does not explain the success of equilibrium statistical mechanics, British Journal for the Philosophy of Science, 47, 63-78 (1996) · Zbl 1133.82300 |
| [11] | Frigg, R., In what sense is the Kolmogorov-Sinai entropy a measure for random behaviour? Bridging the gap between dynamical systems theory and information theory, British Journal for the Philosophy of Science, 55, 411-434 (2004) · Zbl 1077.94507 |
| [12] | Goldstein, H., Classical mechanics (1980), Addison and Wesley: Addison and Wesley Reading/MA · Zbl 0491.70001 |
| [13] | Kelley, J. L., General topology. Graduate Texts in Mathematics, Vol. 27 (1975), Springer: Springer New York · Zbl 0306.54002 |
| [14] | Kim, J., Causation, nomic subsumption and the concept of event, Journal of Philosophy, 70, 217-236 (1973) |
| [15] | Kolmogorov, A. N., On conservation of conditionally periodic motions for a small change in Hamilton’s function, Doklady Akademii Nauk SSSR (N.S.), 98, 527-530 (1954), (Russian) · Zbl 0056.31502 |
| [16] | Krylov, N. S., The processes of relaxation of statistical systems and the criterion of mechanical instability. Thesis 1942, (Works on the Foundations of statistical physics (1979), Princeton University Press: Princeton University Press Princeton, NJ), 193-238 |
| [17] | Lichtenberg, A. J.; Liebermann, M. A., Regular and chaotic dynamics (1992), Springer: Springer Berlin and New York · Zbl 0748.70001 |
| [18] | Mañé, R., Ergodic theory and differentiable dynamics (1983), Springer: Springer Berlin and New York |
| [19] | Markus, L., & Meyer, K. R. (1974). Generic Hamiltonian dynamical systems are neither integrable nor ergodic. Memoirs of the American mathematical society; Markus, L., & Meyer, K. R. (1974). Generic Hamiltonian dynamical systems are neither integrable nor ergodic. Memoirs of the American mathematical society · Zbl 0291.58009 |
| [20] | Moser, J., On invariant curves of area-preserving mappings of an annulus, Nachrichten der Akademie der Wissenschaften in Göttingen: II. Mathematisch-Physikalische Klasse, 1-20 (1962) · Zbl 0107.29301 |
| [21] | Moser, J., Stable and random motions in dynamical systems (1973), Yale University Press: Yale University Press New Haven · Zbl 0271.70009 |
| [22] | Nadkarni, M. G., Basic ergodic theory (1998), Birkhäuser: Birkhäuser Basel · Zbl 0908.28014 |
| [23] | Ornstein, D. S., Ergodic theory, randomness, and dynamical systems (1974), Yale University Press: Yale University Press New Haven · Zbl 0296.28016 |
| [24] | Ornstein, D. S., Ergodic theory, randomness, and ‘chaos’, Science, 243, 182-187 (1989) · Zbl 1226.37018 |
| [25] | Ornstein, D. S.; Weiss, B., Statistical properties of chaotic systems, Bulletin of the American Mathematical Society (New Series), 24, 11-115 (1991) · Zbl 0718.58038 |
| [26] | Ott, E., Chaos in dynamical systems (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0792.58014 |
| [27] | Oxtoby, J. C., Measure and category. A survey of the analogies between topological and measure spaces. Graduate Texts in Mathematics, Vol. 2 (1971), Springer: Springer New York and Berlin · Zbl 0217.09201 |
| [28] | Oxtoby, J. C.; Ulam, S. M., Measure-preserving homeomorphisms and metrical transitivity, Annals of Mathematics (Second Series), 42, 874-920 (1941) · Zbl 0063.06074 |
| [29] | Parry, W., Topics in ergodic theory (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0449.28016 |
| [30] | Petersen, K., Ergodic theory (1983), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0507.28010 |
| [31] | Reichl, L. E., The transition to chaos in conservative systems: Quantum manifestations (1992), Springer: Springer New York and Berlin · Zbl 0776.70003 |
| [32] | Shields, P., The theory of Bernoulli shifts (1973), Chicago University Press: Chicago University Press Chicago · Zbl 0308.28011 |
| [33] | Sinai, Y. G., Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards’, Uspekhi Matematicheskikh Nauk, 25, 2, 152, 141-192 (1970), (Russian) · Zbl 0252.58005 |
| [34] | Sinai, Y. G., On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanicstranslated as Soviet Mathematics Doklady, 4, 1818-1822, Doklady Akademii Nauk SSSR, 153, 1261-1264 (1963), (Russian) · Zbl 07894227 |
| [35] | (Sinai, Y. G., Dynamical systems II. Ergodic theory with applications to dynamical systems and statistical mechanics (1980), Springer: Springer Berlin and Heidelberg) |
| [36] | Simányi, N., The K-property of \(N\) billiard balls, Inventiones Mathematicae, 108, 521-548 (1992) · Zbl 0759.58029 |
| [37] | Simányi, N., Ergodicity of hard spheres in a box, Ergodic Theory and Dynamical Systems, 19, 741-766 (1999) · Zbl 0959.37007 |
| [38] | Simányi, N., Hard ball systems and semi-dispersive billiards: Hyperbolicity and ergodicity. Hard ball systems and the Lorentz gas’, (Encyclopaedia of Mathematical Sciences, Vol. 101 (2000), Springer: Springer Berlin), 51-88 · Zbl 0984.37008 |
| [39] | Simányi, N.; Szász, D., Hard ball systems are completely hyperbolic, Annals of Mathematics (Second Series), 149, 35-96 (1999) · Zbl 0918.58040 |
| [40] | Szász, D., Boltzmann’s ergodic hypothesis, a conjecture for centuries?, Studia Scientiarum Mathematicarum Hungarica, 31, 299-322 (1996) · Zbl 0852.58060 |
| [41] | (Szász, D., Hard Ball Systems and the Lorentz gas. Encyclopaedia of Mathematical Sciences, 101. Mathematical Physics, II (2000), Springer: Springer Berlin) · Zbl 0953.00014 |
| [42] | Tabor, M., Chaos and integrability in nonlinear dynamics. An introduction (1989), Wiley: Wiley New York · Zbl 0682.58003 |
| [43] | Walters, P., An introduction to ergodic theory (1982), Springer: Springer New York · Zbl 0475.28009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
