2020 Ian Snook Prize Problem : Three Routes to the Information Dimensions for One-Dimensional Stochastic Random Walks and Their Equivalent Two-Dimensional Baker Maps

• Autorzy: Hoover W. G. , Hoover C. G.

Identyfikatory

DOI

10.12921/cmst.2019.0000045

• Języki publikacji: EN

Abstrakty

EN

The $1000 Ian Snook Prize for 2020 will be awarded to the author(s) of the most interesting paper exploring pairs of relatively simple, but fractal, models of nonequilibrium systems, dissipative time-reversible Baker Maps and their equivalent stochastic random walks. Two-dimensional deterministic, time-reversible, chaotic, fractal, and dissipative Baker maps are equivalent to stochastic one-dimensional random walks. Three distinct estimates for the information dimension, {0.7897, 0.7415, 0.7337} have all been put forward for one such model. So far there is no cogent explanation for the differences among these estimates. We describe the three routes to the information dimension, DI : 1) iterated Cantor-like mappings, 2) mesh-based analyses of single-point iterations, and 3) the Kaplan-Yorke Lyapunov dimension, thought by many to be exact for these models. We encourage colleagues to address this Prize Problem by suggesting, testing, and analyzing mechanisms underlying these differing results.

Słowa kluczowe

EN

random walks; fractals; Baker Maps; information dimensions; Snook Prize

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2019
• Tom: Vol. 25, No. 4
• Strony: 153--159
• Opis fizyczny: Bibliogr. 14 poz., rys.

Twórcy

autor

Hoover W. G.

• Ruby Valley Research Institute 601 Highway Contract 60 Ruby Valley, Nevada 89833, USA

autor

Hoover C. G.

• Ruby Valley Research Institute 601 Highway Contract 60 Ruby Valley, Nevada 89833, USA

Bibliografia

• [1] Wm.G. Hoover, H.A. Posch, Chaos and Irreversibility in Simple Model Systems, Chaos 8, 366–373 (1998).
• [2] T. Tél, P. Gaspard, G. Nicolis, Proceedings of “Chaos and Irreversibility” at Eötvös University 31.08–06.09.1997, Chaos 8, 309–461 (1998).
• [3] B.L. Holian, W.G. Hoover, H.A. Posch, Resolution of Loschmidt’s Paradox: The Origin of Irreversible Behavior in Reversible Atomistic Dynamics, Physical Review Letters 59, 10–13 (1987).
• [4] W.G. Hoover, H.A. Posch, B.L. Holian, M.J. Gillan, M. Mareschal, C. Massobrio, Dissipative Irreversibility from Nosé’s Reversible Mechanics, Molecular Simulation 1, 79–86 (1987).
• [5] J.D. Farmer, Information Dimension and the Probabilistic Structure of Chaos, Zeitschrift für Naturforschung 37a, 1304–1325 (1982).
• [6] J.D. Farmer, E. Ott, J.A. Yorke, The Dimension of Chaotic Attractors, Physics 7 D, 153–180 (1983).
• [7] W.G. Hoover, C.G. Hoover, Aspects of Dynamical Simulations, Emphasizing Nosé and Nosé-Hoover Dynamics and the Compressible Baker Map, Computational Methods in Science and Technology 25, 125–141 (2019).
• [8] W.G. Hoover, C.G. Hoover, Random Walk Equivalence to the Compressible Baker Map and the Kaplan-Yorke Approximation to Its Information Dimension, arXiv:1909.04526 (2019).
• [9] W.G. Hoover, C.G. Hoover, Microscopic and Macroscopic Simulation Techniques (Kharagpur Lectures), p. 286, World Scientific, Singapore (2018).
• [10] J. Kumicák, ˆ Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005).
• [11] J.L. Kaplan, J.A. Yorke, Chaotic Behavior of Multidimensional Difference Equations, [In:] Functional Differential Equations and the Approximation of Fixed Points, 204–227, ed. by H.O. Peitgen, H.O. Walther, Springer, Berlin (1979).
• [12] C. Grebogi, E. Ott, J.A. Yorke, Roundoff-Induced Periodicity and the Correlation Dimension of Chaotic Attractors, Physical Review A 38, 3688–3692 (1988).
• [13] C. Dellago, Wm.G. Hoover, Finite-Precision Stationary States At and Away from Equilibrium, Physical Review E 62, 6275–6281 (2000). See the references to previous 1988 and 1998 work of Grebogi, Lanford, Ott, and Yorke therein.
• [14] W.G. Hoover, C.G. Tull (now Hoover), H.A. Posch, Negative Lyapunov Exponents for Dissipative Systems, Physics Letters A 131, 211–215 (1988).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).