Time-Symmetry Breaking in Hamiltonian Mechanics. Part III. A Memoir for Douglas James Henderson [1934–2020]
Hoover William G. *, Hoover Carol G.
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833
*E-mail: hooverwilliam@yahoo.com
Received:
Received: 21 November 2020; accepted: 1 December 2020; published online: 10 December 2020
DOI: 10.12921/cmst.2020.0000034
Abstract:
Following Berni Alder [1] and Francis Ree [2], Douglas Henderson was the third of Bill’s California coworkers from the 1960s to die in 2020 [1, 2]. Motivated by Doug’s death we undertook better to understand Lyapunov instability and the breaking of time symmetry in continuum and atomistic simulations. Here we have chosen to extend our explorations of an interesting pair of nonequilibrium systems, the steady shockwave and the unsteady rarefaction wave. We eliminate the need for boundary potentials by simulating the collisions of pairs of mirror-images projectiles. The resulting shock and rarefaction structures are respectively the results of the compression and the expansion of simple fluids. Shockwaves resulting from compression have a steady structure while the rarefaction fans resulting from free expansions continually broaden. We model these processes using classical molecular dynamics and Eulerian fluid mechanics in two dimensions. Although molecular dynamics is time-reversible the reversed simulation of a steady shockwave compression soon results in an unsteady rarefaction fan, violating the microscopic time symmetry of the motion equations but in agreement with the predictions of macroscopic Navier-Stokes fluid mechanics. The explanations for these results are an interesting combination of two (irreversible) instabilities, Lyapunov and Navier-Stokes.
Key words:
irreversibility, Lyapunov instability, shock and rarefaction waves, symmetry breaking
References:
[1] Wm.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics. Part II. A Memoir for Berni Julian Alder [1925–2020], Computational Methods in Science and Technology 26, 101–110 (2020).
[2] Wm.G. Hoover, Compressible Baker Maps and Their Inverses. A Memoir for Francis Hayin Ree [1936–2020], Computational Methods in Science and Technology 26, 5–13 (2020).
[3] Wm.G. Hoover, M. Ross, K.W. Johnson, D. Henderson, J.A. Barker, B.C. Brown, Soft-Sphere Equation of State, Journal of Chemical Physics 52, 4931–4941 (1970).
[4] D.J. Henderson, [In:] Latter-Day Saint Scholars Testify on the FairMormon website (January 2011).
[5] R.P. Feynman, The Relation of Science and Religion, CalTech YMCA talk, 2 May 1956.
[6] R. Dawkins, The Greatest Show on Earth – the Evidence for Evolution, Free Press, New York (2009).
[7] Lj. Milanovic´, H.A. Posch, Wm.G. Hoover, What is ‘Liquid’? Understanding the States of Matter, Molecular Physics 95, 281–287 (1998).
[8] J.A. Barker, D. Henderson, What is ‘Liquid’? Understanding the States of Matter, Reviews of Modern Physics 48, 587–671 (1976).
[9] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[10] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, Journal of Chemical Physics 81, 511–519 (1984).
[11] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[12] L. Boltzmann, Lectures on Gas Theory, transl. by S.G. Brush, Dover, New York (1995).
[13] Wm.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics, Computational Methods in Science and Technology 19, 77–87 (2013).
[14] Wm.G. Hoover, C.G. Hoover, What is Liquid? Lyapunov Instability Reveals Symmetry-Breaking Irreversibilities Hidden Within Hamilton’s Many-Body Equations of Motion, Condensed Matter Physics 18, 1–13 (2015).
[15] Wm.G. Hoover, C.G. Hoover, J.F. Lutsko, Microscopic and Macroscopic Stress with Gravitational and Rotational Forces, Physical Review E 79, 0367098 (2009).
[16] Wm.G. Hoover, Structure of a Shockwave Front in a Liquid, Physical Review Letters 42, 1531–1534 (1979).
[17] M. Ross, B. Alder, Shock Compression of Argon II. Nonadditive Repulsive Potential, Journal of Chemical Physics 46, 4203–4210 (1967).
[18] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, New York (1959).
[19] Wm.G. Hoover, C.G. Hoover, Shockwaves and Local Hydrodynamics; Failure of the Navier-Stokes Equations, Chapter 2 [In:] New Trends in Statistical Physics; Festschrift in Honor of Leopoldo García-Colín’s 80th Birthday, World Scientific, Singapore (2011).
[20] Wm.G. Hoover, C.G. Hoover, Microscopic and Macroscopic Simulation Techniques – Kharagpur Lectures, World Scientific, Singapore (2018).
[21] Wm.G. Hoover, C.G. Hoover, Nonequilibrium Temperature and Thermometry in Heat-Conducting φ4 Models, Physical Review E 77, 041104 (2008).
[22] 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).
Following Berni Alder [1] and Francis Ree [2], Douglas Henderson was the third of Bill’s California coworkers from the 1960s to die in 2020 [1, 2]. Motivated by Doug’s death we undertook better to understand Lyapunov instability and the breaking of time symmetry in continuum and atomistic simulations. Here we have chosen to extend our explorations of an interesting pair of nonequilibrium systems, the steady shockwave and the unsteady rarefaction wave. We eliminate the need for boundary potentials by simulating the collisions of pairs of mirror-images projectiles. The resulting shock and rarefaction structures are respectively the results of the compression and the expansion of simple fluids. Shockwaves resulting from compression have a steady structure while the rarefaction fans resulting from free expansions continually broaden. We model these processes using classical molecular dynamics and Eulerian fluid mechanics in two dimensions. Although molecular dynamics is time-reversible the reversed simulation of a steady shockwave compression soon results in an unsteady rarefaction fan, violating the microscopic time symmetry of the motion equations but in agreement with the predictions of macroscopic Navier-Stokes fluid mechanics. The explanations for these results are an interesting combination of two (irreversible) instabilities, Lyapunov and Navier-Stokes.
Key words:
irreversibility, Lyapunov instability, shock and rarefaction waves, symmetry breaking
References:
[1] Wm.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics. Part II. A Memoir for Berni Julian Alder [1925–2020], Computational Methods in Science and Technology 26, 101–110 (2020).
[2] Wm.G. Hoover, Compressible Baker Maps and Their Inverses. A Memoir for Francis Hayin Ree [1936–2020], Computational Methods in Science and Technology 26, 5–13 (2020).
[3] Wm.G. Hoover, M. Ross, K.W. Johnson, D. Henderson, J.A. Barker, B.C. Brown, Soft-Sphere Equation of State, Journal of Chemical Physics 52, 4931–4941 (1970).
[4] D.J. Henderson, [In:] Latter-Day Saint Scholars Testify on the FairMormon website (January 2011).
[5] R.P. Feynman, The Relation of Science and Religion, CalTech YMCA talk, 2 May 1956.
[6] R. Dawkins, The Greatest Show on Earth – the Evidence for Evolution, Free Press, New York (2009).
[7] Lj. Milanovic´, H.A. Posch, Wm.G. Hoover, What is ‘Liquid’? Understanding the States of Matter, Molecular Physics 95, 281–287 (1998).
[8] J.A. Barker, D. Henderson, What is ‘Liquid’? Understanding the States of Matter, Reviews of Modern Physics 48, 587–671 (1976).
[9] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[10] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, Journal of Chemical Physics 81, 511–519 (1984).
[11] Wm.G. Hoover, Canonical Dynamics: Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[12] L. Boltzmann, Lectures on Gas Theory, transl. by S.G. Brush, Dover, New York (1995).
[13] Wm.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics, Computational Methods in Science and Technology 19, 77–87 (2013).
[14] Wm.G. Hoover, C.G. Hoover, What is Liquid? Lyapunov Instability Reveals Symmetry-Breaking Irreversibilities Hidden Within Hamilton’s Many-Body Equations of Motion, Condensed Matter Physics 18, 1–13 (2015).
[15] Wm.G. Hoover, C.G. Hoover, J.F. Lutsko, Microscopic and Macroscopic Stress with Gravitational and Rotational Forces, Physical Review E 79, 0367098 (2009).
[16] Wm.G. Hoover, Structure of a Shockwave Front in a Liquid, Physical Review Letters 42, 1531–1534 (1979).
[17] M. Ross, B. Alder, Shock Compression of Argon II. Nonadditive Repulsive Potential, Journal of Chemical Physics 46, 4203–4210 (1967).
[18] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, New York (1959).
[19] Wm.G. Hoover, C.G. Hoover, Shockwaves and Local Hydrodynamics; Failure of the Navier-Stokes Equations, Chapter 2 [In:] New Trends in Statistical Physics; Festschrift in Honor of Leopoldo García-Colín’s 80th Birthday, World Scientific, Singapore (2011).
[20] Wm.G. Hoover, C.G. Hoover, Microscopic and Macroscopic Simulation Techniques – Kharagpur Lectures, World Scientific, Singapore (2018).
[21] Wm.G. Hoover, C.G. Hoover, Nonequilibrium Temperature and Thermometry in Heat-Conducting φ4 Models, Physical Review E 77, 041104 (2008).
[22] 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).
