A Quarter Century of Baker-Map Exploration
Hoover William G. 1, Hoover Carol G. 2
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833, USA
1 E-mail: hooverwilliam@yahoo.com
2 E-mail: hoover1carol@yahoo.com
Received:
Received: 9 April 2023; revised: 17 April 2023; accepted: 18 April 2023; published online: 17 May 2023
DOI: 10.12921/cmst.2023.0000007
Abstract:
25 years ago the June 1998 Focus Issue of “Chaos” described the proceedings of a workshop meeting held in Budapest and called “Chaos and Irreversibility”, by the organizers, T. Tél, P. Gaspard, and G. Nicolis. These editors organized the meeting and the proceedings’ issue. They emphasized the importance of fractal structures and Lyapunov instability to modelling nonequilibrium steady states. Several papers concerning maps were presented. Ronald Fox considered the entropy of the incompressible Baker Map B(x, y), shown here in Fig. 1. He found that the limiting probability density after many applications of the map is ambiguous, depending upon the way the limit is approached. Harald Posch and Bill Hoover considered a time-reversible version of a compressible Baker Map, with the compressibility modelling thermostatting. Now, 25 years later, we have uncovered a similar ambiguity, with the information dimension of the probability density giving one value from pointwise averaging and a different one with areawise averaging. Goldstein, Lebowitz, and Sinai appear to consider similar ambiguities. Tasaki, Gilbert, and Dorfman note that the Baker Map probability density is singular everywhere, though integrable over the fractal y coordinate. Breymann, Tél, and Vollmer considered the concatenation of Baker Maps into MultiBaker Maps, as a step toward measuring spatial transport with dynamical systems. The present authors have worked on Baker Maps ever since the 1997 Budapest meeting described in “Chaos”. This paper provides a number of computational benchmark simulations of “Generalized Baker Maps” (where the compressibility of the Map is varied or “generalized”) as described by Kumicˆák in 2005.
Key words:
Baker Map, fractal, information dimension, Kaplan-Yorke dimension, random walk
References:
[1] 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).
[2] B. Moran, W.G. Hoover, S. Bestiale, Diffusion in a Peri- odic Lorentz Gas, Journal of Statistical Physics 48, 709–726 (1987).
[3] 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).
[4] Wm.G. Hoover, H.A. Posch, Chaos and Irreversibility in Simple Model Systems, [In:] Proceedings of “Chaos and Ir- reversibility” at Eötvös University 31 August–6 September (1997), organized by T. Te´l, P. Gaspard, G. Nicolis, Chaos 8, 366–374 (1998).
[5] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative At- tractors In Three And Four Phase-Space Dimensions, Physi- cal Review E 55, 6803–6810 (1997).
[6] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[7] S. Nosé, A Unified Formulation of the Constant Temper- ature Molecular Dynamics Methods, Journal of Chemical Physics 81, 511–519 (1984).
[8] W.G. Hoover, Canonical Dynamics: Equilibrium Phase- Space Distributions, Physical Review A 31, 1695–1697 (1985).
[9] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Physical Review A 33, 4253–4265 (1986).
[10] W.G. Hoover, B. Moran, Phase-Space Singularities in Atomistic Planar Diffusive Flow, Physical Review A 40, 5319–5326 (1989).
[11] E. Ott, W.D. Withers, J.A. Yorke, Is the Dimension of Chaotic Attractors Invariant Under Coordinate Changes?, The Jour- nal of Statistical Physics 36, 687–697 (1984).
[12] T. Tél, M. Gruiz, Chaotic Dynamics; An Introduction Based on Classical Mechanics, Cambridge University Press (2006).
[13] J.L. Kaplan, J.A. Yorke, Chaotic Behavior of Multidi- mensional Difference Equations, [In:] Functional Differ- ential Equations and the Approximation of Fixed Points, Eds. H.O. Peitgen, H.O. Walther, Springer, Berlin, 204–227 (1979).
[14] T. Tél, P. Gaspard, G. Nicolis, Chaos and Irreversibility: In- troductory Comments, Chaos 8, 309 (1988).
[15] R.J. Fox, Construction of the Jordan Basis for the Baker Map, Chaos 7, 254–269 (1997).
[16] J. Kumicˆák, Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005).
[17] W.G. Hoover, C.G. Hoover, F. Grond, Phase-Space Growth Rates, Local Lyapunov Spectra, and Symmetry Breaking for Time-Reversible Dissipative Oscillators, Communications in Nonlinear Science and Numerical Simulation 13, 1180–1193 (2006).
[18] W.G. Hoover, Compressible Baker Maps and Their Inverses. A Memoir for Francis Hayin Ree [1936–2020], Computa- tional Methods in Science and Technology 26, 5–13 (2020).
[19] J.D. Farmer, Information Dimension and the Probabilis- tic Structure of Chaos, Zeitschrift für Naturforschung 3A, 1304–1325 (1982).
[20] S. Tasaki, T. Gilbert, J.R. Dorfman, An Analytical Construc- tion of the SRB Measures for Baker-Type Maps, Chaos 8, 424–443 (1998). Note Fig. 4.
25 years ago the June 1998 Focus Issue of “Chaos” described the proceedings of a workshop meeting held in Budapest and called “Chaos and Irreversibility”, by the organizers, T. Tél, P. Gaspard, and G. Nicolis. These editors organized the meeting and the proceedings’ issue. They emphasized the importance of fractal structures and Lyapunov instability to modelling nonequilibrium steady states. Several papers concerning maps were presented. Ronald Fox considered the entropy of the incompressible Baker Map B(x, y), shown here in Fig. 1. He found that the limiting probability density after many applications of the map is ambiguous, depending upon the way the limit is approached. Harald Posch and Bill Hoover considered a time-reversible version of a compressible Baker Map, with the compressibility modelling thermostatting. Now, 25 years later, we have uncovered a similar ambiguity, with the information dimension of the probability density giving one value from pointwise averaging and a different one with areawise averaging. Goldstein, Lebowitz, and Sinai appear to consider similar ambiguities. Tasaki, Gilbert, and Dorfman note that the Baker Map probability density is singular everywhere, though integrable over the fractal y coordinate. Breymann, Tél, and Vollmer considered the concatenation of Baker Maps into MultiBaker Maps, as a step toward measuring spatial transport with dynamical systems. The present authors have worked on Baker Maps ever since the 1997 Budapest meeting described in “Chaos”. This paper provides a number of computational benchmark simulations of “Generalized Baker Maps” (where the compressibility of the Map is varied or “generalized”) as described by Kumicˆák in 2005.
Key words:
Baker Map, fractal, information dimension, Kaplan-Yorke dimension, random walk
References:
[1] 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).
[2] B. Moran, W.G. Hoover, S. Bestiale, Diffusion in a Peri- odic Lorentz Gas, Journal of Statistical Physics 48, 709–726 (1987).
[3] 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).
[4] Wm.G. Hoover, H.A. Posch, Chaos and Irreversibility in Simple Model Systems, [In:] Proceedings of “Chaos and Ir- reversibility” at Eötvös University 31 August–6 September (1997), organized by T. Te´l, P. Gaspard, G. Nicolis, Chaos 8, 366–374 (1998).
[5] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative At- tractors In Three And Four Phase-Space Dimensions, Physi- cal Review E 55, 6803–6810 (1997).
[6] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[7] S. Nosé, A Unified Formulation of the Constant Temper- ature Molecular Dynamics Methods, Journal of Chemical Physics 81, 511–519 (1984).
[8] W.G. Hoover, Canonical Dynamics: Equilibrium Phase- Space Distributions, Physical Review A 31, 1695–1697 (1985).
[9] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Physical Review A 33, 4253–4265 (1986).
[10] W.G. Hoover, B. Moran, Phase-Space Singularities in Atomistic Planar Diffusive Flow, Physical Review A 40, 5319–5326 (1989).
[11] E. Ott, W.D. Withers, J.A. Yorke, Is the Dimension of Chaotic Attractors Invariant Under Coordinate Changes?, The Jour- nal of Statistical Physics 36, 687–697 (1984).
[12] T. Tél, M. Gruiz, Chaotic Dynamics; An Introduction Based on Classical Mechanics, Cambridge University Press (2006).
[13] J.L. Kaplan, J.A. Yorke, Chaotic Behavior of Multidi- mensional Difference Equations, [In:] Functional Differ- ential Equations and the Approximation of Fixed Points, Eds. H.O. Peitgen, H.O. Walther, Springer, Berlin, 204–227 (1979).
[14] T. Tél, P. Gaspard, G. Nicolis, Chaos and Irreversibility: In- troductory Comments, Chaos 8, 309 (1988).
[15] R.J. Fox, Construction of the Jordan Basis for the Baker Map, Chaos 7, 254–269 (1997).
[16] J. Kumicˆák, Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005).
[17] W.G. Hoover, C.G. Hoover, F. Grond, Phase-Space Growth Rates, Local Lyapunov Spectra, and Symmetry Breaking for Time-Reversible Dissipative Oscillators, Communications in Nonlinear Science and Numerical Simulation 13, 1180–1193 (2006).
[18] W.G. Hoover, Compressible Baker Maps and Their Inverses. A Memoir for Francis Hayin Ree [1936–2020], Computa- tional Methods in Science and Technology 26, 5–13 (2020).
[19] J.D. Farmer, Information Dimension and the Probabilis- tic Structure of Chaos, Zeitschrift für Naturforschung 3A, 1304–1325 (1982).
[20] S. Tasaki, T. Gilbert, J.R. Dorfman, An Analytical Construc- tion of the SRB Measures for Baker-Type Maps, Chaos 8, 424–443 (1998). Note Fig. 4.
