2024 Snook Prize Winner: Mixing Rates of Ergodic Algorithms
Hoover William G., Hoover Carol G.
850 Ruby Vista Drive, Unit 321
Elko, Nevada 89801, USA
E-mail: hooverwilliam@yahoo.com
E-mail: hoover1carol@yahoo.com
Received:
Received: 14 November 2025; accepted: 18 November 2025
DOI: 10.12921/cmst.2025.0000023
Abstract:
The Snook Prizes (2014–2024) were stimulated by Shuichi Nosé’s IBM–Japan prize award for time-reversible dynamics generating the canonical distribution [1]. The 2024 Snook Prize has been awarded to Clint Sprott for his exploration of six thermostatted harmonic oscillators at unit temperature that produce ergodic Gaussian distributions of position q and momentum p. The simplest example of a bang-bang controller with a signum nonlinearity reaches equilibrium most quickly as measured by the time-dependent kurtosis, Kp(t) = ⟨p4⟩/⟨p2⟩2.
Key words:
ergodicity, Gibbs’ canonical distribution, mixing, Snook Prize
References:
[1] Wm.G. Hoover, Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize, CMST 20, 87–92 (2014).
[2] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase Space Distributions, Physical Review A 31, 1695–1697 (1985).
[3] P.K. Patra, Wm.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 (2016).
[4] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat, CMST 23, 11–18 (2017).
[5] Wm.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[6] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155–185 and 214, 180–218 (1992).
[7] 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).
[8] J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, CMST 24, 169–176 (2018).
[9] Wm.G. Hoover, C.G. Hoover, 2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates, CMST 29, 65–69 (2023).
[10] J.C. Sprott, Mixing Rates of Ergodic Algorithms,
CMST 30, 5–9 (2024).
The Snook Prizes (2014–2024) were stimulated by Shuichi Nosé’s IBM–Japan prize award for time-reversible dynamics generating the canonical distribution [1]. The 2024 Snook Prize has been awarded to Clint Sprott for his exploration of six thermostatted harmonic oscillators at unit temperature that produce ergodic Gaussian distributions of position q and momentum p. The simplest example of a bang-bang controller with a signum nonlinearity reaches equilibrium most quickly as measured by the time-dependent kurtosis, Kp(t) = ⟨p4⟩/⟨p2⟩2.
Key words:
ergodicity, Gibbs’ canonical distribution, mixing, Snook Prize
References:
[1] Wm.G. Hoover, Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize, CMST 20, 87–92 (2014).
[2] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase Space Distributions, Physical Review A 31, 1695–1697 (1985).
[3] P.K. Patra, Wm.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 (2016).
[4] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat, CMST 23, 11–18 (2017).
[5] Wm.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[6] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155–185 and 214, 180–218 (1992).
[7] 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).
[8] J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, CMST 24, 169–176 (2018).
[9] Wm.G. Hoover, C.G. Hoover, 2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates, CMST 29, 65–69 (2023).
[10] J.C. Sprott, Mixing Rates of Ergodic Algorithms,
CMST 30, 5–9 (2024).
