Ergodicity of the Martyna-Klein-Tuckerman Thermostat and the 2014 Ian Snook Prize
Ruby Valley Research Institute, Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
Received: 27 January 2015; accepted: 30 January 2015; published online: 05 March 2015
DOI: 10.12921/cmst.2015.21.01.002
Abstract:
Nosé and Hoover’s 1984 work showed that although Nosé and Nosé-Hoover dynamics were both consistent with Gibbs’ canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs’ Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient, to generate Gibbs’ canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of two-thermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tuckerman in 1992. Its ergodicity was called into question by Patra and Bhattacharya in 2014. This question became the subject of the 2014 Snook Prize. Here we summarize the previous work on this problem and elucidate new details of the chaotic dynamics in the neighborhood of the two fixed points. We apply six separate tests for ergodicity and conclude that the MKT equations are fully compatible with all of them, in consonance with our recent work with Clint Sprott and Puneet Patra.
Key words:
Chaotic Dynamics, ergodicity, fixed points, time reversibility
References:
[1] S. Nosé, “A Unified Formulation of the Constant Tempera-
ture Molecular Dynamics Methods”, The Journal of Chemical
Physics 81, 511-519 (1984).
[2] S. Nosé, “A Molecular Dynamics Method for Simulations
in the Canonical Ensemble”, Molecular Physics 52, 191-198
(1984).
[3] W. G. Hoover, “Canonical Dynamics: Equilibrium Phase-
Space Distributions”, Physical Review A 31, 1695-1697
(1985).
[4] W. G. Hoover and B. L. Holian, “Kinetic Moments Method
for the Canonical Ensemble Distribution”, Physics Letters A
211, 253-257 (1996).
[5] N. Ju and A. Bulgac, “Finite-Temperature Properties of
Sodium Clusters”, Physical Review B 48, 2721-2732 (1993).
[6] G. J. Martyna, M. L. Klein, and M. Tuckerman, “Nosé-Hoover
Chains: the Canonical Ensemble via Continuous Dynamics”,
The Journal of Chemical Physics 97, 2635-2643 (1992).
[7] D. Kusnezov, A. Bulgac and W. Bauer, “Canonical Ensembles
from Chaos”, Annals of Physics (NY) 204, 155-185 (1990).
[8] A. Bulgac and D. Kusnezov, “Canonical Ensemble Averages
from Pseudomicrocanonical Dynamics”, Physical Review A
42, 5045-5048 (1990).
[9] P. K. Patra and B. Bhattacharya, “Non-Ergodicity of Nosé-
Hoover Chain Thermostat in Computationally Achievable
Time”, Physical Review E 90, 043304 (2014) = arχiv:
1407.2353 (2014).
[10] Wm. G. Hoover and C. G. Hoover, “Ergodicity of a Time-
Reversibly Thermostated Harmonic Oscillator and the 2014
Ian Snook Prize”, Computational Methods in Science and
Technology 20, 87-92 (2014).
[11] W. G. Hoover, J. C. Sprott, P. K. Patra, and C. G. Hoover, “De-
terministic Time-Reversible Thermostats: Chaos, Ergodicity,
and the Zeroth Law of Thermodynamics”, arχiv 1501.03875
(2015), Molecular Physics (in press).
[12] H. A. Posch, W. G. Hoover, and F. J. Vesely, “Canonical Dy-
namics of the Nosé Oscillator: Stability, Order, and Chaos”,
Physical Review A 33, 4253-4265 (1986).
[13] W. G. Hoover and H. A. Posch, “Direct Measurement of Equi-
librium and Nonequilibrium Lyapunov Spectra”, Physics Let-
ters A 123, 227-230 (1987).
Nosé and Hoover’s 1984 work showed that although Nosé and Nosé-Hoover dynamics were both consistent with Gibbs’ canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs’ Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient, to generate Gibbs’ canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of two-thermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tuckerman in 1992. Its ergodicity was called into question by Patra and Bhattacharya in 2014. This question became the subject of the 2014 Snook Prize. Here we summarize the previous work on this problem and elucidate new details of the chaotic dynamics in the neighborhood of the two fixed points. We apply six separate tests for ergodicity and conclude that the MKT equations are fully compatible with all of them, in consonance with our recent work with Clint Sprott and Puneet Patra.
Key words:
Chaotic Dynamics, ergodicity, fixed points, time reversibility
References:
[1] S. Nosé, “A Unified Formulation of the Constant Tempera-
ture Molecular Dynamics Methods”, The Journal of Chemical
Physics 81, 511-519 (1984).
[2] S. Nosé, “A Molecular Dynamics Method for Simulations
in the Canonical Ensemble”, Molecular Physics 52, 191-198
(1984).
[3] W. G. Hoover, “Canonical Dynamics: Equilibrium Phase-
Space Distributions”, Physical Review A 31, 1695-1697
(1985).
[4] W. G. Hoover and B. L. Holian, “Kinetic Moments Method
for the Canonical Ensemble Distribution”, Physics Letters A
211, 253-257 (1996).
[5] N. Ju and A. Bulgac, “Finite-Temperature Properties of
Sodium Clusters”, Physical Review B 48, 2721-2732 (1993).
[6] G. J. Martyna, M. L. Klein, and M. Tuckerman, “Nosé-Hoover
Chains: the Canonical Ensemble via Continuous Dynamics”,
The Journal of Chemical Physics 97, 2635-2643 (1992).
[7] D. Kusnezov, A. Bulgac and W. Bauer, “Canonical Ensembles
from Chaos”, Annals of Physics (NY) 204, 155-185 (1990).
[8] A. Bulgac and D. Kusnezov, “Canonical Ensemble Averages
from Pseudomicrocanonical Dynamics”, Physical Review A
42, 5045-5048 (1990).
[9] P. K. Patra and B. Bhattacharya, “Non-Ergodicity of Nosé-
Hoover Chain Thermostat in Computationally Achievable
Time”, Physical Review E 90, 043304 (2014) = arχiv:
1407.2353 (2014).
[10] Wm. G. Hoover and C. G. Hoover, “Ergodicity of a Time-
Reversibly Thermostated Harmonic Oscillator and the 2014
Ian Snook Prize”, Computational Methods in Science and
Technology 20, 87-92 (2014).
[11] W. G. Hoover, J. C. Sprott, P. K. Patra, and C. G. Hoover, “De-
terministic Time-Reversible Thermostats: Chaos, Ergodicity,
and the Zeroth Law of Thermodynamics”, arχiv 1501.03875
(2015), Molecular Physics (in press).
[12] H. A. Posch, W. G. Hoover, and F. J. Vesely, “Canonical Dy-
namics of the Nosé Oscillator: Stability, Order, and Chaos”,
Physical Review A 33, 4253-4265 (1986).
[13] W. G. Hoover and H. A. Posch, “Direct Measurement of Equi-
librium and Nonequilibrium Lyapunov Spectra”, Physics Let-
ters A 123, 227-230 (1987).
