Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize
Ruby Valley Research Institute Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
Received: 18 July 2016; accepted: 01 August 2016; published online: 04 August 2016
DOI: 10.12921/cmst.2016.0000037
Abstract:
For a harmonic oscillator, Nosé’s single-thermostat approach to simulating Gibbs’ canonical ensemble with dynamics samples only a small fraction of the phase space. Nosé’s approach has been improved in a series of three steps: [1] several two-thermostat sets of motion equations have been found which cover the complete phase space in an ergodic fashion; [2] sets of single-thermostat motion equations, exerting “weak control” over both forces and momenta, have been shown to be ergodic; and [3] sets of single-thermostat motion equations exerting weak control over two velocity moments provide ergodic phase-space sampling for the oscillator and for the rigid pendulum, but not for the quartic oscillator or for the Mexican Hat potential. The missing fourth step, motion equations providing ergodic sampling for anharmonic potentials requires a further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of the most interesting original submission addressing the problem of finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat.
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 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] 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).
[5] P.K. Patra, W.G. Hoover, C.G. Hoover, and J.C. Sprott, The
Equivalence of Dissipation from Gibbs’ Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Os- cillators, International Journal of Bifurcation and Chaos 26, 1650089 (2016).
[6] Wm.G. Hoover and C.G. Hoover, Hamiltonian Dynamics of Thermostated Systems: Two-Temperature Heat-Conducting φ4 Chains, Journal of Chemical Physics 126, 164113 (2007).
[7] Wm.G. Hoover and C.G. Hoover, Hamiltonian Thermostats Fail to Promote Heat Flow, Communications in Nonlinear Science and Numerical Simulation 18, 3365-3372 (2013).
[8] A.C. Brańka, M. Kowalik, and K.W. Wojciechowski, Generalization of the Nosé-Hoover Approach, The Journal of Chemical Physics 119, 1929-1936 (2003).
[9] D. Kusnezov, A. Bulgac, and W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155-185 (1990).
[10] D. Kusnezov and A. Bulgac, Canonical Ensembles from Chaos: Constrained Dynamical Systems, Annals of Physics 214, 180-218 (1992).
[11] Wm. G. Hoover and B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253-257 (1996).
[12] P.K. Patra and B. Bhattacharya, A Deterministic Thermostat for Controlling Temperature Using All Degrees of Freedom, The Journal of Chemical Physics 140, 064106 (2014).
[13] A.Sergi and G.S. Ezra, Bulgac-Kusnezov-Nosé-HooverThermostats, Physical Review E 81, 036705 (2010), Figure 2.
[14] W.G. Hoover, J.C. Sprott, and C.G. Hoover, Ergodicity of a Singly-Thermostated Harmonic Oscillator, Communications in Nonlinear Science and Numerical Simulation 32, 234-240 (2016).
[15] W.G. Hoover, C.G. Hoover, and J.C. Sprott, Nonequilibrium Systems: Hard Disks and Harmonic Oscillators Near and Far From Equilibrium, Molecular Simulation (in press).
[16] A.Sergi and M.Ferrario, Non-Hamiltonian Equations of Motion with a Conserved Energy, Physical Review E 64, 056125 (2001), Equations 24, 26, and 27. See Reference 18.
[17] 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).
[18] W.G. Hoover, J.C. Sprott, and P.K. Patra, Ergodic Time-Reversible Chaos for Gibbs’ Canonical Oscillator, Physics Letters A 379, 2395-2400 (2015).
For a harmonic oscillator, Nosé’s single-thermostat approach to simulating Gibbs’ canonical ensemble with dynamics samples only a small fraction of the phase space. Nosé’s approach has been improved in a series of three steps: [1] several two-thermostat sets of motion equations have been found which cover the complete phase space in an ergodic fashion; [2] sets of single-thermostat motion equations, exerting “weak control” over both forces and momenta, have been shown to be ergodic; and [3] sets of single-thermostat motion equations exerting weak control over two velocity moments provide ergodic phase-space sampling for the oscillator and for the rigid pendulum, but not for the quartic oscillator or for the Mexican Hat potential. The missing fourth step, motion equations providing ergodic sampling for anharmonic potentials requires a further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of the most interesting original submission addressing the problem of finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat.
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 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] 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).
[5] P.K. Patra, W.G. Hoover, C.G. Hoover, and J.C. Sprott, The
Equivalence of Dissipation from Gibbs’ Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Os- cillators, International Journal of Bifurcation and Chaos 26, 1650089 (2016).
[6] Wm.G. Hoover and C.G. Hoover, Hamiltonian Dynamics of Thermostated Systems: Two-Temperature Heat-Conducting φ4 Chains, Journal of Chemical Physics 126, 164113 (2007).
[7] Wm.G. Hoover and C.G. Hoover, Hamiltonian Thermostats Fail to Promote Heat Flow, Communications in Nonlinear Science and Numerical Simulation 18, 3365-3372 (2013).
[8] A.C. Brańka, M. Kowalik, and K.W. Wojciechowski, Generalization of the Nosé-Hoover Approach, The Journal of Chemical Physics 119, 1929-1936 (2003).
[9] D. Kusnezov, A. Bulgac, and W. Bauer, Canonical Ensembles from Chaos, Annals of Physics 204, 155-185 (1990).
[10] D. Kusnezov and A. Bulgac, Canonical Ensembles from Chaos: Constrained Dynamical Systems, Annals of Physics 214, 180-218 (1992).
[11] Wm. G. Hoover and B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253-257 (1996).
[12] P.K. Patra and B. Bhattacharya, A Deterministic Thermostat for Controlling Temperature Using All Degrees of Freedom, The Journal of Chemical Physics 140, 064106 (2014).
[13] A.Sergi and G.S. Ezra, Bulgac-Kusnezov-Nosé-HooverThermostats, Physical Review E 81, 036705 (2010), Figure 2.
[14] W.G. Hoover, J.C. Sprott, and C.G. Hoover, Ergodicity of a Singly-Thermostated Harmonic Oscillator, Communications in Nonlinear Science and Numerical Simulation 32, 234-240 (2016).
[15] W.G. Hoover, C.G. Hoover, and J.C. Sprott, Nonequilibrium Systems: Hard Disks and Harmonic Oscillators Near and Far From Equilibrium, Molecular Simulation (in press).
[16] A.Sergi and M.Ferrario, Non-Hamiltonian Equations of Motion with a Conserved Energy, Physical Review E 64, 056125 (2001), Equations 24, 26, and 27. See Reference 18.
[17] 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).
[18] W.G. Hoover, J.C. Sprott, and P.K. Patra, Ergodic Time-Reversible Chaos for Gibbs’ Canonical Oscillator, Physics Letters A 379, 2395-2400 (2015).
