Canonical Temperature Control by Molecular Dynamics
Hoover William G. 1, Hoover Carol G. 2
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833, USA
1 E-mail: hooverwilliam@yahoo.com
2 E-mail: hoover1carol@yahoo.com
Received:
Received: 8 February 2024; in final form: 15 February 2024; accepted: 16 February 2024; published online: 3 March 2024
DOI: 10.12921/cmst.2024.0000005
Abstract:
“Pedagogical derivations for Nosé’s dynamics can be developed in two different ways, (i) by starting with a temperature-dependent Hamiltonian in which the variable s scales the time or the mass, or (ii) by requiring that the equations of motion generate the canonical distribution including a Gaussian distribution in the friction coefficient ζ. Nosé’s papers follow the former approach. Because the latter approach is not only constructive and simple, but also can be generalized to other forms of the equations of motion, we illustrate it here. We begin by considering the probability density f (q, p, ζ) in an extended phase space which includes ζ as well as all pairs of phase variables q and p. This density f (q, p, ζ) satisfies the conservation of probability (Liouville’s Continuity Equation)
(∂f/∂t) + Sum[(∂(q˙f )/∂q)] + Sum[(∂(p˙f )/∂p)] + Sum[(∂(ζ˙f )/∂ζ)] = 0 .”
The multi-authored “review” [1] motivated our quoting the history of Nosé and Nosé-Hoover mechanics, aptly described on page 31 of Bill’s 1986 Molecular Dynamics book, reproduced above [2].
Key words:
cell model, continuity equation, Nosé and Nosé-Hoover mechanics
References:
[1] C. Massobrio, I.A. Essomba, M. Boero, C. Diarra, M. Guerboub, K. Ishisone, A. Lambrecht, E. Martin, I. Morrot-Woisard, G. Ori, C. Tugène, S.D.W. Wendji, On the Actual Difference Between the Nosé and the Nosé-Hoover Thermostats: A Critical Review of Canonical Temperature Control by Molecular Dynamics, Physica Status Solidi B 261, 2300209 (2023).
[2] Wm.G. Hoover, Molecular Dynamics, Springer-Verlag, Berlin (1986).
[3] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Method, The Journal of Chemical Physics 81, 511–519 (1984).
[4] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[5] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[6] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Physical Review A 33, 4253–4265 (1986).
[7] W.G. Hoover, C.G. Hoover, Microscopic and Macroscopic Simulation Techniques, World Scientific, Singapore (2018).
[8] C.P. Dettmann, G.P. Morriss, Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nosé-Hoover Dynamics, Physical Review E 55, 3693–3696 (1996).
[9] C.P. Dettmann, G.P. Morriss, Hamiltonian Formulation of the Gaussian Isokinetic Thermostat, Physical Review E 54, 2495–2500 (1996).
[10] D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, ANU Press, Canberra (2007).
[11] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos II: Constrained Dynamical Systems, Annals of Physics 214, 180–218 (1992).
[12] W.G. Hoover, H.A. Posch, B.L. Holian, M.J. Gillan, M. Mareschal, C. Massobrio, Dissipative Irreversibility from Nosé’s Reversible Mechanics, Molecular Simulation 1, 79–86 (1987).
[13] B.L. Holian, W.G. Hoover, H.A. Posch, Resolution of Loschmidt’s Paradox: The Origin of Irreversible Behavior in Reversible Atomistic Dynamics, Physical Review Letters 59, 10–13 (1987).
[14] J.C. Sprott, W.G. Hoover, C.G. Hoover, Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators with a Temperature Gradient, Physical Review E 89, 042914 (2014).
[15] L. Wang, X-S. Yang, A Vast Amount of Various Invariant Tori in the Nosé-Hoover Oscillator, Chaos 25, 123110 (2015).
[16] L. Wang, X-S. Yang, The Invariant Tori of Knot Type and the Interlinked Invariant Tori in the Nosé-Hoover System, The European Physical Journal B 88, 78 (2015).
“Pedagogical derivations for Nosé’s dynamics can be developed in two different ways, (i) by starting with a temperature-dependent Hamiltonian in which the variable s scales the time or the mass, or (ii) by requiring that the equations of motion generate the canonical distribution including a Gaussian distribution in the friction coefficient ζ. Nosé’s papers follow the former approach. Because the latter approach is not only constructive and simple, but also can be generalized to other forms of the equations of motion, we illustrate it here. We begin by considering the probability density f (q, p, ζ) in an extended phase space which includes ζ as well as all pairs of phase variables q and p. This density f (q, p, ζ) satisfies the conservation of probability (Liouville’s Continuity Equation)
(∂f/∂t) + Sum[(∂(q˙f )/∂q)] + Sum[(∂(p˙f )/∂p)] + Sum[(∂(ζ˙f )/∂ζ)] = 0 .”
The multi-authored “review” [1] motivated our quoting the history of Nosé and Nosé-Hoover mechanics, aptly described on page 31 of Bill’s 1986 Molecular Dynamics book, reproduced above [2].
Key words:
cell model, continuity equation, Nosé and Nosé-Hoover mechanics
References:
[1] C. Massobrio, I.A. Essomba, M. Boero, C. Diarra, M. Guerboub, K. Ishisone, A. Lambrecht, E. Martin, I. Morrot-Woisard, G. Ori, C. Tugène, S.D.W. Wendji, On the Actual Difference Between the Nosé and the Nosé-Hoover Thermostats: A Critical Review of Canonical Temperature Control by Molecular Dynamics, Physica Status Solidi B 261, 2300209 (2023).
[2] Wm.G. Hoover, Molecular Dynamics, Springer-Verlag, Berlin (1986).
[3] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Method, The Journal of Chemical Physics 81, 511–519 (1984).
[4] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[5] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[6] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Physical Review A 33, 4253–4265 (1986).
[7] W.G. Hoover, C.G. Hoover, Microscopic and Macroscopic Simulation Techniques, World Scientific, Singapore (2018).
[8] C.P. Dettmann, G.P. Morriss, Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nosé-Hoover Dynamics, Physical Review E 55, 3693–3696 (1996).
[9] C.P. Dettmann, G.P. Morriss, Hamiltonian Formulation of the Gaussian Isokinetic Thermostat, Physical Review E 54, 2495–2500 (1996).
[10] D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, ANU Press, Canberra (2007).
[11] D. Kusnezov, A. Bulgac, Canonical Ensembles from Chaos II: Constrained Dynamical Systems, Annals of Physics 214, 180–218 (1992).
[12] W.G. Hoover, H.A. Posch, B.L. Holian, M.J. Gillan, M. Mareschal, C. Massobrio, Dissipative Irreversibility from Nosé’s Reversible Mechanics, Molecular Simulation 1, 79–86 (1987).
[13] B.L. Holian, W.G. Hoover, H.A. Posch, Resolution of Loschmidt’s Paradox: The Origin of Irreversible Behavior in Reversible Atomistic Dynamics, Physical Review Letters 59, 10–13 (1987).
[14] J.C. Sprott, W.G. Hoover, C.G. Hoover, Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nosé-Hoover Oscillators with a Temperature Gradient, Physical Review E 89, 042914 (2014).
[15] L. Wang, X-S. Yang, A Vast Amount of Various Invariant Tori in the Nosé-Hoover Oscillator, Chaos 25, 123110 (2015).
[16] L. Wang, X-S. Yang, The Invariant Tori of Knot Type and the Interlinked Invariant Tori in the Nosé-Hoover System, The European Physical Journal B 88, 78 (2015).
