Aspects of Dynamical Simulations, Emphasizing Nosé and Nosé-Hoover Dynamics and the Compressible Baker Map
Hoover William G. *, Hoover Carol G.
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833
*E-mail: hooverwilliam@yahoo.com
Received:
Received: 12 August 2019; accepted: 20 August 2019; published online: 23 September 2019
DOI: 10.12921/cmst.2019.0000035
Abstract:
Aspects of the Nosé and Nosé-Hoover dynamics developed in 1983–1984 along with Dettmann’s closely related dynamics of 1996, are considered. We emphasize paradoxes associated with Liouville’s Theorem. Our account is pedagogical, focused on the harmonic oscillator for simplicity, though exactly the same ideas can be, and have been, applied to manybody systems. Nosé, Nosé-Hoover, and Dettmann flows were all developed in order to access Gibbs’ canonical ensemble directly from molecular dynamics. Unlike Monte Carlo algorithms dynamical flow models are often not ergodic and so can fail to reproduce Gibbs’ ensembles. Accordingly we include a discussion of ergodicity, the visiting of all relevant microstates corresponding to the desired ensemble. We consider Lyapunov instability too, the usual mechanism for phasespace mixing. We show that thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, depending upon the chosen “phase space”. The fractal nature of nonequilibrium flows is also illustrated for two simple two-dimensional models, the hard-disk-based Galton Board and the time-reversible Baker Map. The simultaneous treatment of flows as one-dimensional and many-dimensional suggests some interesting topological problems for future investigations.
Key words:
Baker Map, Dettmann oscillator, Galton Board, nonlinear dynamics, Nosé oscillator, Nosé-Hoover oscillator
References:
[1] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics 21, 1087–1092 (1953).
[2] B.J. Alder, T.E. Wainwright, Molecular Motions, Scientific American 201, 113–126 (1959).
[3] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, The Journal of Chemical Physics 81, 511–519 (1984).
[4] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[5] W.G. Hoover, Canonical Dynamics. Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[6] 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).
[7] W.G. Hoover, Mécanique de Nonéquilibre à la Californienne, Physica A 240, 1–11 (1997).
[8] S.D. Bond, B.J. Leimkuhler, B.B. Laird, The Nosé-Poincaré Method for Constant Temperature Molecular Dynamics, Journal of Computational Physics 151, 114–134 (1999).
[9] C.P. Dettmann, G.P. Morriss, Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nosé-Hoover Dynamics, Physical Review E 55, 3693–3696 (1997).
[10] S. Nosé, An Improved Symplectic Integrator for Nosé-Poincaré Thermostat, Journal of the Physical Society of Japan 70, 75–77 (2001).
[11] J.C. Sprott, Some Simple Chaotic Flows, Physical Review E, 50, 647–650 (1994).
[12] W.G. Hoover, Remark on “Some Simple Chaotic Flows”, Physical Review E 51, 759–760 (1995).
[13] W.G. Hoover, Molecular Dynamics, [In:] Lecture Notes in Physics #258, Springer-Verlag, Berlin (1986). A scanned copy of the book is available free at williamhoover.info.
[14] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155–185 (1990) and 214, 180–218 (1992).
[15] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat, Computational Methods in Science and Technology 23, 11–18 (2017).
[16] J.C. Sprott, Variants of the Nosé-Hoover Oscillator, European Physics Journal Special Topics Issue “Special Chaotic Systems” (preprint, July 2019) following up J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, Computational Methods in Science and Technology 24, 169–176 (2018).
[17] W.G. Hoover, J.C. Sprott, C.G. Hoover, A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems: Comparing the Nosé and Nosé-Hoover Oscillator Dynamics, American Journal of Physics 84, 786 (2016).
[18] J. Ford, The Fermi-Pasta-Ulam Problem: Paradox Turns Discovery, Physics Reports 213, 271–310 (1992).
[19] B.J. Alder, T.E. Wainwright, Phase Transition for a Hard Sphere System, The Journal of Chemical Physics 27, 1208–1209 (1957).
[20] W.W. Wood, J.D. Jacobson, Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres, The Journal of Chemical Physics 27, 1207–1208 (1957).
[21] Wm.G. Hoover, Computational Statistical Mechanics, [In:] Studies in Modern Thermodynamics #10, Elsevier, Amsterdam (1991). A scanned copy of the book is available free on the web at williamhoover.info.
[22] M.A.M. Karlsen, The Early Years of Molecular Dynamics and Computers at UCRL, LRL, LLL, and LLNL, [In:] Symposium in Honor of Dr Berni Alder’s 90th Birthday, ed. by E. Schwegler, B.M. Rubenstein, S.B. Libby, Advances in the Computational Sciences, World Scientific, Singapore (2017).
[23] M. Engel, J.A. Anderson, S.C. Glotzer, M. Isobe, E.P. Bernard, W. Krauth, Hard-Disk Equation of State: First-Order Liquid-Hexatic Transition in Two Dimensions with Three Simulation Methods, Physical Review 87, 042134 (2013).
[24] B. Moran, W.G. Hoover, S. Bestiale, Diffusion in a Periodic Lorentz Gas, Journal of Statistical Physics 48, 709–726 (1987).
[25] 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).
[26] B.L. Holian, W.G. Hoover, B. Moran, G.K. Straub, Shockwave Structure via Nonequilibrium Molecular Dynamics and Navier-Stokes Continuum Mechanics, Physical Review A 22, 2798–2808 (1980).
[27] J.C. Sprott, W.G. Hoover, C.G. Hoover, Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators with a Temperature Gradient, Physical Review E 89, 042914 (2014).
[28] L. Wang, X.S. Yang, The Coexistence of Invariant Tori and Topological Horseshoes in a Generalized Nosé-Hoover Oscillator, International Journal of Bifurcation and Chaos 27, 1750111 (2017).
[29] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[30] Wm.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 (2008).
[31] W.G. Hoover, C.G. Hoover, J.C. Sprott, Nonequilibrium Systems: Hard Disks and Harmonic Oscillators Near and Far From Equilibrium, Molecular Simulation 42, 1300–1316 (2016).
[32] J.L. Kaplan, J.A. Yorke, Chaotic Behavior of Multidimensional Difference Equations, [In:] Functional Differential Equations and the Approximation of Fixed Points, ed. by H.O. Peitgen, H.O. Walther, 204–227, Springer, Berlin (1979).
[33] J.D. Farmer, E. Ott, J.A. Yorke, The Dimension of Chaotic Attractors, Physica D, 7, 153–180 (1983).
[34] Wm.G. Hoover, H.A. Posch, Chaos and Irreversibility in Simple Model Systems, Chaos 8, 366–373 (1998).
[35] J. Kumicák, Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005).
[36] W.G. Hoover, C.G. Tull (now Hoover), H.A. Posch, Negative Lyapunov Exponents for Dissipative Systems, Physics Letters A 131, 211–215 (1988).
[37] Wm.G. Hoover, C.G. Hoover, H.A. Posch, J.A. Codelli, The Second Law of Thermodynamics and Multifractal Distribution Functions: Bin Counting, Pair Correlations, and [the Definite Failure of] the Kaplan-Yorke Conjecture, Communications in Nonlinear Science and Numerical Simulation 12, 214–231 (2005).
[38] 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 1998 work of Grebogi, Lanford, Ott, and Yorke therein.
[39] Wm.G. Hoover, H.A. Posch, K. Aoki, D. Kusnezov, Remarks on Non-Hamiltonian Statistical Mechanics: Lyapunov Exponents and Phase-Space Dimensionality Loss, Europhysics Letters 60, 337–341 (2002).
Aspects of the Nosé and Nosé-Hoover dynamics developed in 1983–1984 along with Dettmann’s closely related dynamics of 1996, are considered. We emphasize paradoxes associated with Liouville’s Theorem. Our account is pedagogical, focused on the harmonic oscillator for simplicity, though exactly the same ideas can be, and have been, applied to manybody systems. Nosé, Nosé-Hoover, and Dettmann flows were all developed in order to access Gibbs’ canonical ensemble directly from molecular dynamics. Unlike Monte Carlo algorithms dynamical flow models are often not ergodic and so can fail to reproduce Gibbs’ ensembles. Accordingly we include a discussion of ergodicity, the visiting of all relevant microstates corresponding to the desired ensemble. We consider Lyapunov instability too, the usual mechanism for phasespace mixing. We show that thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, depending upon the chosen “phase space”. The fractal nature of nonequilibrium flows is also illustrated for two simple two-dimensional models, the hard-disk-based Galton Board and the time-reversible Baker Map. The simultaneous treatment of flows as one-dimensional and many-dimensional suggests some interesting topological problems for future investigations.
Key words:
Baker Map, Dettmann oscillator, Galton Board, nonlinear dynamics, Nosé oscillator, Nosé-Hoover oscillator
References:
[1] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics 21, 1087–1092 (1953).
[2] B.J. Alder, T.E. Wainwright, Molecular Motions, Scientific American 201, 113–126 (1959).
[3] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, The Journal of Chemical Physics 81, 511–519 (1984).
[4] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[5] W.G. Hoover, Canonical Dynamics. Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[6] 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).
[7] W.G. Hoover, Mécanique de Nonéquilibre à la Californienne, Physica A 240, 1–11 (1997).
[8] S.D. Bond, B.J. Leimkuhler, B.B. Laird, The Nosé-Poincaré Method for Constant Temperature Molecular Dynamics, Journal of Computational Physics 151, 114–134 (1999).
[9] C.P. Dettmann, G.P. Morriss, Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nosé-Hoover Dynamics, Physical Review E 55, 3693–3696 (1997).
[10] S. Nosé, An Improved Symplectic Integrator for Nosé-Poincaré Thermostat, Journal of the Physical Society of Japan 70, 75–77 (2001).
[11] J.C. Sprott, Some Simple Chaotic Flows, Physical Review E, 50, 647–650 (1994).
[12] W.G. Hoover, Remark on “Some Simple Chaotic Flows”, Physical Review E 51, 759–760 (1995).
[13] W.G. Hoover, Molecular Dynamics, [In:] Lecture Notes in Physics #258, Springer-Verlag, Berlin (1986). A scanned copy of the book is available free at williamhoover.info.
[14] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155–185 (1990) and 214, 180–218 (1992).
[15] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat, Computational Methods in Science and Technology 23, 11–18 (2017).
[16] J.C. Sprott, Variants of the Nosé-Hoover Oscillator, European Physics Journal Special Topics Issue “Special Chaotic Systems” (preprint, July 2019) following up J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, Computational Methods in Science and Technology 24, 169–176 (2018).
[17] W.G. Hoover, J.C. Sprott, C.G. Hoover, A Tutorial: Adaptive Runge-Kutta Integration for Stiff Systems: Comparing the Nosé and Nosé-Hoover Oscillator Dynamics, American Journal of Physics 84, 786 (2016).
[18] J. Ford, The Fermi-Pasta-Ulam Problem: Paradox Turns Discovery, Physics Reports 213, 271–310 (1992).
[19] B.J. Alder, T.E. Wainwright, Phase Transition for a Hard Sphere System, The Journal of Chemical Physics 27, 1208–1209 (1957).
[20] W.W. Wood, J.D. Jacobson, Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres, The Journal of Chemical Physics 27, 1207–1208 (1957).
[21] Wm.G. Hoover, Computational Statistical Mechanics, [In:] Studies in Modern Thermodynamics #10, Elsevier, Amsterdam (1991). A scanned copy of the book is available free on the web at williamhoover.info.
[22] M.A.M. Karlsen, The Early Years of Molecular Dynamics and Computers at UCRL, LRL, LLL, and LLNL, [In:] Symposium in Honor of Dr Berni Alder’s 90th Birthday, ed. by E. Schwegler, B.M. Rubenstein, S.B. Libby, Advances in the Computational Sciences, World Scientific, Singapore (2017).
[23] M. Engel, J.A. Anderson, S.C. Glotzer, M. Isobe, E.P. Bernard, W. Krauth, Hard-Disk Equation of State: First-Order Liquid-Hexatic Transition in Two Dimensions with Three Simulation Methods, Physical Review 87, 042134 (2013).
[24] B. Moran, W.G. Hoover, S. Bestiale, Diffusion in a Periodic Lorentz Gas, Journal of Statistical Physics 48, 709–726 (1987).
[25] 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).
[26] B.L. Holian, W.G. Hoover, B. Moran, G.K. Straub, Shockwave Structure via Nonequilibrium Molecular Dynamics and Navier-Stokes Continuum Mechanics, Physical Review A 22, 2798–2808 (1980).
[27] J.C. Sprott, W.G. Hoover, C.G. Hoover, Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators with a Temperature Gradient, Physical Review E 89, 042914 (2014).
[28] L. Wang, X.S. Yang, The Coexistence of Invariant Tori and Topological Horseshoes in a Generalized Nosé-Hoover Oscillator, International Journal of Bifurcation and Chaos 27, 1750111 (2017).
[29] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[30] Wm.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 (2008).
[31] W.G. Hoover, C.G. Hoover, J.C. Sprott, Nonequilibrium Systems: Hard Disks and Harmonic Oscillators Near and Far From Equilibrium, Molecular Simulation 42, 1300–1316 (2016).
[32] J.L. Kaplan, J.A. Yorke, Chaotic Behavior of Multidimensional Difference Equations, [In:] Functional Differential Equations and the Approximation of Fixed Points, ed. by H.O. Peitgen, H.O. Walther, 204–227, Springer, Berlin (1979).
[33] J.D. Farmer, E. Ott, J.A. Yorke, The Dimension of Chaotic Attractors, Physica D, 7, 153–180 (1983).
[34] Wm.G. Hoover, H.A. Posch, Chaos and Irreversibility in Simple Model Systems, Chaos 8, 366–373 (1998).
[35] J. Kumicák, Irreversibility in a Simple Reversible Model, Physical Review E 71, 016115 (2005).
[36] W.G. Hoover, C.G. Tull (now Hoover), H.A. Posch, Negative Lyapunov Exponents for Dissipative Systems, Physics Letters A 131, 211–215 (1988).
[37] Wm.G. Hoover, C.G. Hoover, H.A. Posch, J.A. Codelli, The Second Law of Thermodynamics and Multifractal Distribution Functions: Bin Counting, Pair Correlations, and [the Definite Failure of] the Kaplan-Yorke Conjecture, Communications in Nonlinear Science and Numerical Simulation 12, 214–231 (2005).
[38] 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 1998 work of Grebogi, Lanford, Ott, and Yorke therein.
[39] Wm.G. Hoover, H.A. Posch, K. Aoki, D. Kusnezov, Remarks on Non-Hamiltonian Statistical Mechanics: Lyapunov Exponents and Phase-Space Dimensionality Loss, Europhysics Letters 60, 337–341 (2002).
