Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator 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: 12 July 2014; accepted: 16 July 2014; published online: 18 July 2014
DOI: 10.12921/cmst.2014.20.03.87-92
Abstract:
Shuichi Nosé opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs’ canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling. Nosé’s dynamical Hamiltonian bridge requires the “ergodic” support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nosé’s dynamical approach failed to agree with Gibbs’ sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated “thermostated” equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive (contradictory) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.
Key words:
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 Meth-
ods, 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] C.P. Dettmann and G.P. Morriss, Hamiltonian Formulation of
the Gaussian Isokinetic Thermostat, Physical Review E 54,
2495-2500 (1996).
[5] 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).
[6] H.A. Posch and Wm.G. Hoover, Time-Reversible Dissipative
Attractors in Three and Four Phase-Space Dimensions, Phys-
ical Review E 55, 6803-6810 (1997).
[7] D. Kusnezov, A. Bulgac, and W. Bauer, Canonical Ensembles
from Chaos, Annals of Physics 204, 155-185 (1990).
[8] D. Kusnezov and A. Bulgac, Canonical Ensembles from
Chaos: Constrained Dynamical Systems, Annals of Physics
214, 180-218 (1992).
[9] G.J. Martyna, M.L. Klein, and M. Tuckerman, Nosé-Hoover
Chains – the Canonical Ensemble via Continuous Dynamics,
Journal of Chemical Physics 97, 2635-2643 (1992).
[10] Wm.G. Hoover and B.L. Holian, Kinetic Moments Method
for the Canonical Ensemble Distribution, Physics Letters A
211, 253-257 (1996).
[11] P.K. Patra and B. Bhattacharya, A Deterministic Thermostat
for Controlling Temperature using All Degrees of Freedom,
Journal of Chemical Physics 140, 064106 (2014).
[12] K.P. Travis and C. Braga, Configurational Temperature and
Pressure Molecular Dynamics: Review of Current Methodo-
logy and Applications to the Shear Flow of a Simple Fluid,
Molecular Physics 104, 3735-3749 (2006).
[13] Wm.G. Hoover, C.G. Hoover, and D. Isbister, Chaos, Ergodic
Convergence, and Fractal Instability for a Thermostated
Canonical Harmonic Oscillator, Physical Review E 63,
3541-3546 (2000).
[14] Wm.G. Hoover and C.G. Hoover, Time-Reversible Random
Number Generators: Solution of Our Challenge by Federico
Ricci-Tersenghi: arXiv.1305.0961.
[15] P.K. Patra and B. Bhattacharya, Non-Ergodicity of Nosé-
Hoover Chain Thermostat in Computationally Achievable
Time: arXiv.1407.2353.
[16] F. Ricci-Tersenghi, The Solution to the Challenge in ‘Time-
Reversible Random Number Generators’ by Wm. G. Hoover
and Carol G. Hoover: arXiv.1305.1805.
Shuichi Nosé opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs’ canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling. Nosé’s dynamical Hamiltonian bridge requires the “ergodic” support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nosé’s dynamical approach failed to agree with Gibbs’ sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated “thermostated” equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive (contradictory) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.
Key words:
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 Meth-
ods, 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] C.P. Dettmann and G.P. Morriss, Hamiltonian Formulation of
the Gaussian Isokinetic Thermostat, Physical Review E 54,
2495-2500 (1996).
[5] 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).
[6] H.A. Posch and Wm.G. Hoover, Time-Reversible Dissipative
Attractors in Three and Four Phase-Space Dimensions, Phys-
ical Review E 55, 6803-6810 (1997).
[7] D. Kusnezov, A. Bulgac, and W. Bauer, Canonical Ensembles
from Chaos, Annals of Physics 204, 155-185 (1990).
[8] D. Kusnezov and A. Bulgac, Canonical Ensembles from
Chaos: Constrained Dynamical Systems, Annals of Physics
214, 180-218 (1992).
[9] G.J. Martyna, M.L. Klein, and M. Tuckerman, Nosé-Hoover
Chains – the Canonical Ensemble via Continuous Dynamics,
Journal of Chemical Physics 97, 2635-2643 (1992).
[10] Wm.G. Hoover and B.L. Holian, Kinetic Moments Method
for the Canonical Ensemble Distribution, Physics Letters A
211, 253-257 (1996).
[11] P.K. Patra and B. Bhattacharya, A Deterministic Thermostat
for Controlling Temperature using All Degrees of Freedom,
Journal of Chemical Physics 140, 064106 (2014).
[12] K.P. Travis and C. Braga, Configurational Temperature and
Pressure Molecular Dynamics: Review of Current Methodo-
logy and Applications to the Shear Flow of a Simple Fluid,
Molecular Physics 104, 3735-3749 (2006).
[13] Wm.G. Hoover, C.G. Hoover, and D. Isbister, Chaos, Ergodic
Convergence, and Fractal Instability for a Thermostated
Canonical Harmonic Oscillator, Physical Review E 63,
3541-3546 (2000).
[14] Wm.G. Hoover and C.G. Hoover, Time-Reversible Random
Number Generators: Solution of Our Challenge by Federico
Ricci-Tersenghi: arXiv.1305.0961.
[15] P.K. Patra and B. Bhattacharya, Non-Ergodicity of Nosé-
Hoover Chain Thermostat in Computationally Achievable
Time: arXiv.1407.2353.
[16] F. Ricci-Tersenghi, The Solution to the Challenge in ‘Time-
Reversible Random Number Generators’ by Wm. G. Hoover
and Carol G. Hoover: arXiv.1305.1805.
