2020 Ian Snook Prize Problem: Three Routes to the Information Dimensions for One-Dimensional Stochastic Random Walks and Their Equivalent Two-Dimensional Baker Maps
Hoover William G. *, Hoover Carol G.
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833, USA
*E-mail: hooverwilliam@yahoo.com
Received:
Received: 29 October 2019; revised: 11 November 2019; accepted: 11 November 2019; published online: 18 December 2019
DOI: 10.12921/cmst.2019.0000045
Abstract:
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, f0:7897; 0:7415; 0:7337g 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.
Key words:
Baker Maps, fractals, information dimensions, random walks, Snook Prize
References:
[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. Kumiˆcák, 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).
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, f0:7897; 0:7415; 0:7337g 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.
Key words:
Baker Maps, fractals, information dimensions, random walks, Snook Prize
References:
[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. Kumiˆcák, 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).
