2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates
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: 2 September 2023; in final form: 3 September 2023; accepted: 8 September 2023
DOI: 10.12921/cmst.2023.0000022
Abstract:
In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs’ canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion in such a way as to generate the Gaussian velocity distribution ∝ e^(−p2/2mkT) and the corresponding potential distribution, ∝ e^(−Φ(q)/kT) . (For convenience here we choose Boltzmann’s constant k and the particle massmboth equal to unity.) SoonWilliam Hoover pointed out that Nosé’s approach fails for the simple harmonic oscillator [3]. Rather than generating the entire Gaussian canonical oscillator distribution, the Nosé-Hoover approach, which includes an additional friction coefficient ζ with distribution e^(−ζ2/2)/√2π, generates only a modest fractal chaotic sea, filling a small percentage of the canonical (q, p, ζ) distribution. In the decade that followed this thermostatted work a handful of ergodic algorithms were developed in both three- and four-dimensional phase spaces. These new approaches generated the entire canonical distribution, without holes. The 2024 Snook Prize problem is to study the efficiency of several such algorithms, such as the five ergodic examples described here, so as to assess their relative usefulness in attaining the canonical steady state for the harmonic oscillator. The 2024 Prize rewarding the best assessment is United States $1000, half of it a gift from ourselves with the balance from the Pozna´n SuperComputer Center.
Key words:
ergodicity, fractals, Gibbs’ canonical distribution, Lyapunov instability
References:
[1] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Method, The Journal of Chemical Physics 81, 511–519 (1984).
[2] S. Nosé, Constant Temperature Molecular Dynamics Methods, Progress in Theoretical Physics Supplement 103, 1–46 (1991).
[3] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[4] 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).
[5] W.G. Hoover, J.C. Sprott, C.G. Hoover, Adaptive Runge-Kutta Integration for Stiff Systems: Comparing Nosé and Nosé-Hoover Dynamics for the Harmonic Oscillator, American Journal of Physics 84, 786–794 (2016).
[6] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative Attractors in Three and Four Phase-Space Dimensions, Physical Review E 55, 6803–6810 (1997).
[7] P.K. Patra, W.G. Hoover, C.G. Hoover, J.C. Sprott, The Equivalence of Dissipation from Gibbs’ Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators, International Journal of Bifurcation and Chaos 26, 1650089-1–11 (2016).
[8] D. Tapias, A. Bravetti, D. Sanders, Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat, Computational Methods in Science and Technology 23, 11–18 (2017).
[9] J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, Computational Methods in Science and Technology 24, 169–176 (2018).
[10] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[11] A. Bulgac, D. Kusnezov, Canonical Ensemble Averages from Pseudomicrocanonical Dynamics, Physical Review A 42, 5045–5048(R) (1990).
[12] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics (New York) 204, 155–185 (1990).
[13] N.Ju, A. Bulgac, Finite-Temperature Properties of Sodium Clusters, Physical Review B 48, 2721–2732 (1993).
[14] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos II: Constrained Dynamical Systems, Annals of Physics (New York) 214, 180–218 (1992).
[15] G.J. Martyna, M.L. Klein, M. Tuckerman, Nosé-Hoover Chains; the Canonical Ensemble via Continuous Dynamics, The Journal of Chemical Physics 97, 2635–2643 (1992).
[16] W.G. Hoover, C.G. Hoover, K.P. Travis, Information Dimensions of Simple Four-Dimensional Flows, Computational Methods in Science and Technology 29, 21–26 (2023).
In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs’ canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion in such a way as to generate the Gaussian velocity distribution ∝ e^(−p2/2mkT) and the corresponding potential distribution, ∝ e^(−Φ(q)/kT) . (For convenience here we choose Boltzmann’s constant k and the particle massmboth equal to unity.) SoonWilliam Hoover pointed out that Nosé’s approach fails for the simple harmonic oscillator [3]. Rather than generating the entire Gaussian canonical oscillator distribution, the Nosé-Hoover approach, which includes an additional friction coefficient ζ with distribution e^(−ζ2/2)/√2π, generates only a modest fractal chaotic sea, filling a small percentage of the canonical (q, p, ζ) distribution. In the decade that followed this thermostatted work a handful of ergodic algorithms were developed in both three- and four-dimensional phase spaces. These new approaches generated the entire canonical distribution, without holes. The 2024 Snook Prize problem is to study the efficiency of several such algorithms, such as the five ergodic examples described here, so as to assess their relative usefulness in attaining the canonical steady state for the harmonic oscillator. The 2024 Prize rewarding the best assessment is United States $1000, half of it a gift from ourselves with the balance from the Pozna´n SuperComputer Center.
Key words:
ergodicity, fractals, Gibbs’ canonical distribution, Lyapunov instability
References:
[1] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Method, The Journal of Chemical Physics 81, 511–519 (1984).
[2] S. Nosé, Constant Temperature Molecular Dynamics Methods, Progress in Theoretical Physics Supplement 103, 1–46 (1991).
[3] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[4] 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).
[5] W.G. Hoover, J.C. Sprott, C.G. Hoover, Adaptive Runge-Kutta Integration for Stiff Systems: Comparing Nosé and Nosé-Hoover Dynamics for the Harmonic Oscillator, American Journal of Physics 84, 786–794 (2016).
[6] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative Attractors in Three and Four Phase-Space Dimensions, Physical Review E 55, 6803–6810 (1997).
[7] P.K. Patra, W.G. Hoover, C.G. Hoover, J.C. Sprott, The Equivalence of Dissipation from Gibbs’ Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators, International Journal of Bifurcation and Chaos 26, 1650089-1–11 (2016).
[8] D. Tapias, A. Bravetti, D. Sanders, Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat, Computational Methods in Science and Technology 23, 11–18 (2017).
[9] J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, Computational Methods in Science and Technology 24, 169–176 (2018).
[10] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[11] A. Bulgac, D. Kusnezov, Canonical Ensemble Averages from Pseudomicrocanonical Dynamics, Physical Review A 42, 5045–5048(R) (1990).
[12] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics (New York) 204, 155–185 (1990).
[13] N.Ju, A. Bulgac, Finite-Temperature Properties of Sodium Clusters, Physical Review B 48, 2721–2732 (1993).
[14] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos II: Constrained Dynamical Systems, Annals of Physics (New York) 214, 180–218 (1992).
[15] G.J. Martyna, M.L. Klein, M. Tuckerman, Nosé-Hoover Chains; the Canonical Ensemble via Continuous Dynamics, The Journal of Chemical Physics 97, 2635–2643 (1992).
[16] W.G. Hoover, C.G. Hoover, K.P. Travis, Information Dimensions of Simple Four-Dimensional Flows, Computational Methods in Science and Technology 29, 21–26 (2023).
