Mixing Rates of Ergodic Algorithms
University of Wisconsin-Madison
Department of Physics
Madison, Wisconsin 53706, USA
E-mail: sprott@physics.wisc.edu
Received:
Received: 8 January 2024; in final form: 30 January 2024; accepted: 31 January 2024; published online: 19 February 2024
DOI: 10.12921/cmst.2024.0000001
Abstract:
In response to the 2024 Snook Prize Problem, this paper compares the mixing rates of six simple numerical algorithms that produce an ergodic Gaussian distribution of position and momentum for a one-dimensional harmonic oscillator. A hundred thousand initial conditions spread uniformly over the constant energy surface are used for each of the six systems. The time-dependent kurtosis serves as a measure of the mixing rate. By this criterion, the most rapid mixing occurs for the signum thermostat system with an optimally chosen parameter value.
Key words:
ergodicity, Gibbs’ canonical distribution, mixing, Snook Prize
References:
[1] Wm.G. Hoover, C.G. Hoover, 2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates, CMST 29, 65–69 (2023).
[2] J.W. Gibbs, Elementary Princples in Statistical Mechanics, Yale University Press (1902); Reprinted Dover Publications (2014).
[3] J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, CMST 24, 169–176 (2018).
[4] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press (2007).
[5] 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).
[6] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat, CMST 23, 11–18 (2017).
[7] Wm.G. Hoover, C.G. Hoover, Singly-thermostatted Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize, CMST 22, 127–131 (2016).
[8] Wm.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[9] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155–185 (1990).
[10] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos II: Constrained Dynamical Systems, Annals of Physics 214, 180–218 (1992).
[11] 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).
[12] J.C. Sprott, Wm.G. Hoover, C.G. Hoover, Elegant Simulations: From Simple Oscillators to Many-Body Systems, World Scientific (2023).
In response to the 2024 Snook Prize Problem, this paper compares the mixing rates of six simple numerical algorithms that produce an ergodic Gaussian distribution of position and momentum for a one-dimensional harmonic oscillator. A hundred thousand initial conditions spread uniformly over the constant energy surface are used for each of the six systems. The time-dependent kurtosis serves as a measure of the mixing rate. By this criterion, the most rapid mixing occurs for the signum thermostat system with an optimally chosen parameter value.
Key words:
ergodicity, Gibbs’ canonical distribution, mixing, Snook Prize
References:
[1] Wm.G. Hoover, C.G. Hoover, 2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates, CMST 29, 65–69 (2023).
[2] J.W. Gibbs, Elementary Princples in Statistical Mechanics, Yale University Press (1902); Reprinted Dover Publications (2014).
[3] J.C. Sprott, Ergodicity of One-Dimensional Oscillators with a Signum Thermostat, CMST 24, 169–176 (2018).
[4] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press (2007).
[5] 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).
[6] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-dimensional Systems Coupled to the Logistic Thermostat, CMST 23, 11–18 (2017).
[7] Wm.G. Hoover, C.G. Hoover, Singly-thermostatted Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize, CMST 22, 127–131 (2016).
[8] Wm.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[9] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155–185 (1990).
[10] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos II: Constrained Dynamical Systems, Annals of Physics 214, 180–218 (1992).
[11] 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).
[12] J.C. Sprott, Wm.G. Hoover, C.G. Hoover, Elegant Simulations: From Simple Oscillators to Many-Body Systems, World Scientific (2023).
