Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize

• Autorzy: Hoover W. G. , Hoover C. G.

Identyfikatory

DOI

10.12921/cmst.2014.20.03.87-92

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

ergodicity; chaos; algorithms; dynamical systems

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2014
• Tom: Vol. 20, No. 3
• Strony: 87--92
• Opis fizyczny: Bibliogr. 16 poz., rys.

Twórcy

autor

Hoover W. G.

• Ruby Valley Research Institute Highway Contract 60, Box 601 Ruby Valley, Nevada 89833

autor

Hoover C. G.

• Ruby Valley Research Institute Highway Contract 60, Box 601 Ruby Valley, Nevada 89833

Bibliografia

• [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] 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 Dynamics 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, Physical 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 Methodology 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.