• NEWS
  • CURRENT ISSUE
  • CONTACT
GET_pdf

Volume 25 (4) 2019, 153–159

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 Wm.G. *, Hoover C.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).

  • JOURNAL MENU

    • AIMS AND SCOPE
    • EDITORS
    • EDITORIAL BOARD
    • NOTES FOR AUTHORS
    • CONTACT
    • IAN SNOOK PRIZES 2015
    • IAN SNOOK PRIZES 2016
    • IAN SNOOK PRIZES 2017
    • IAN SNOOK PRIZES 2018
    • IAN SNOOK PRIZES 2019
  • GALLERY

    vol_19_04_2013
    vol_19_03_2013
    volume_19_2_2013
    volume_19_1_2013
    vol_sp_2_2010
    vol_sp_2006
    vol_sp_1_2010
    vol_16_01_2010
    vol_18_2_2012
    vol_18_01_2012
    vol_17_01_02_2011
    vol_16_02_2010
    vol_15_02_2009
    vol_15_01_2009
    vol_14_02_2008
    vol_14_01_2008
    vol_13_02_2007
    vol_13_01_2007
    vol_12_02_2006
    vol_12_01_2006
    vol_11_02_2005
    vol_11_01_2005
  • CURRENT ISSUE

  • MANUSCRIPT SUBMISSION

    • SUBMIT A MANUSCRIPT
  • FUTURE ISSUES

    • ACCEPTED PAPERS
  • ALL ISSUES

    • 2020
      • Volume 26 (4)
      • Volume 26 (3)
      • Volume 26 (2)
      • Volume 26 (1)
    • 2019
      • Volume 25 (4)
      • Volume 25 (3)
      • Volume 25 (2)
      • Volume 25 (1)
    • 2018
      • Volume 24 (4)
      • Volume 24 (3)
      • Volume 24 (2)
      • Volume 24 (1)
    • 2017
      • Volume 23 (4)
      • Volume 23 (3)
      • Volume 23 (2)
      • Volume 23 (1)
    • 2016
      • Volume 22 (4)
      • Volume 22 (3)
      • Volume 22 (2)
      • Volume 22 (1)
    • 2015
      • Volume 21 (4)
      • Volume 21 (3)
      • Volume 21 (2)
      • Volume 21 (1)
    • 2014
      • Volume 20 (4)
      • Volume 20 (3)
      • Volume 20 (2)
      • Volume 20 (1)
    • 2013
      • Volume 19 (4)
      • Volume 19 (3)
      • Volume 19 (2)
      • Volume 19 (1)
    • 2012
      • Volume 18 (2)
      • Volume 18 (1)
    • 2011
      • Volume 17 (1-2)
    • 2010
      • Volume SI (2)
      • Volume SI (1)
      • Volume 16 (2)
      • Volume 16 (1)
    • 2009
      • Volume 15 (2)
      • Volume 15 (1)
    • 2008
      • Volume 14 (2)
      • Volume 14 (1)
    • 2007
      • Volume 13 (2)
      • Volume 13 (1)
    • 2006
      • Volume SI (1)
      • Volume 12 (2)
      • Volume 12 (1)
    • 2005
      • Volume 11 (2)
      • Volume 11 (1)
    • 2004
      • Volume 10 (2)
      • Volume 10 (1)
    • 2003
      • Volume 9 (1)
    • 2002
      • Volume 8 (2)
      • Volume 8 (1)
    • 2001
      • Volume 7 (2)
      • Volume 7 (1)
    • 2000
      • Volume 6 (1)
    • 1999
      • Volume 5 (1)
    • 1998
      • Volume 4 (1)
    • 1997
      • Volume 3 (1)
    • 1996
      • Volume 2 (1)
      • Volume 1 (1)
    • OLDER ISSUES
  • DATABASES

    • ARTICLES BASE
    • AUTHORS BASE
  • NEWS
  • CURRENT ISSUE
  • CONTACT

Institute of Bioorganic Chemistry Polish Academy of Sciences
Poznań Supercomputing and Networking Center

61-704 Poznań, Z. Noskowskiego 12/14
phone: (+48 61) 858-20-03
fax: (+48 61) 858-21-51