Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize Award
Ruby Valley Research Institute Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
Received: 20 January 2017; accepted: 20 January 2017; published online: 31 January 2017
DOI: 10.12921/cmst.2017.0000005
Abstract:
The 2016 Snook Prize has been awarded to Diego Tapias, Alessandro Bravetti, and David Sanders for their paper “Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat”. They introduced a relatively-stiff hyperbolic tangent thermostat force and successfully tested its ability to reproduce Gibbs’ canonical distribution for three one-dimensional problems, the harmonic oscillator, the quartic oscillator, and the Mexican Hat potentials: {(q2 /2); (q 4 /4); (q 4 /4) − (q 2 /2)}. Their work constitutes an effective response to the 2016 Ian Snook Prize Award goal, “finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat”. We confirm their work here and highlight an interesting feature of the Mexican Hat problem when it is solved with an adaptive integrator.
Key words:
algorithms, chaos, dynamical systems, ergodicity, Ian Snook Prize
References:
[1] S. Nosé, A Unified Formulation of the Constant Temperature
Molecular Dynamics Methods, 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] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles
from Chaos, Annals of Physics 204, 155–185 (1990).
[5] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos:
Constrained Dynamical Systems, Annals of Physics 214, 180–
218 (1992).
[6] Wm.G. Hoover, B.L. Holian, Kinetic Moments Method for
the Canonical Ensemble Distribution, Physics Letters A 211,
253–257 (1996).
[7] Wm.G. Hoover, C.G. Hoover, Singly-Thermostated Ergodic-
ity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook
Prize, Computational Methods in Science and Technology 22,
127–131 (2016).
[8] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-
Dimensional Systems Coupled to the Logistic Thermostat,
Computational Methods in Science and Technology (in press,
2017) = arXiv 1611.05090.
[9] Wm.G. Hoover, C.G. Hoover, Comparison of Very Smooth
Cell-Model Trajectories Using Five Symplectic and Two
Runge-Kutta Integrators, Computational Methods in Science
and Technology 21, 109–116 (2015).
The 2016 Snook Prize has been awarded to Diego Tapias, Alessandro Bravetti, and David Sanders for their paper “Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat”. They introduced a relatively-stiff hyperbolic tangent thermostat force and successfully tested its ability to reproduce Gibbs’ canonical distribution for three one-dimensional problems, the harmonic oscillator, the quartic oscillator, and the Mexican Hat potentials: {(q2 /2); (q 4 /4); (q 4 /4) − (q 2 /2)}. Their work constitutes an effective response to the 2016 Ian Snook Prize Award goal, “finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat”. We confirm their work here and highlight an interesting feature of the Mexican Hat problem when it is solved with an adaptive integrator.
Key words:
algorithms, chaos, dynamical systems, ergodicity, Ian Snook Prize
References:
[1] S. Nosé, A Unified Formulation of the Constant Temperature
Molecular Dynamics Methods, 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] D. Kusnezov, A. Bulgac, W. Bauer, Canonical Ensembles
from Chaos, Annals of Physics 204, 155–185 (1990).
[5] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos:
Constrained Dynamical Systems, Annals of Physics 214, 180–
218 (1992).
[6] Wm.G. Hoover, B.L. Holian, Kinetic Moments Method for
the Canonical Ensemble Distribution, Physics Letters A 211,
253–257 (1996).
[7] Wm.G. Hoover, C.G. Hoover, Singly-Thermostated Ergodic-
ity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook
Prize, Computational Methods in Science and Technology 22,
127–131 (2016).
[8] D. Tapias, A. Bravetti, D.P. Sanders, Ergodicity of One-
Dimensional Systems Coupled to the Logistic Thermostat,
Computational Methods in Science and Technology (in press,
2017) = arXiv 1611.05090.
[9] Wm.G. Hoover, C.G. Hoover, Comparison of Very Smooth
Cell-Model Trajectories Using Five Symplectic and Two
Runge-Kutta Integrators, Computational Methods in Science
and Technology 21, 109–116 (2015).
