Time-Reversible Thermodynamic Irreversibility: One-Dimensional Heat-Conducting Oscillators and Two-Dimensional Newtonian Shockwaves
Hoover William G. *, Hoover Carol G.
Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833, USA∗E-mail: hooverwilliam@yahoo.com
Received:
Received: 1 October 2022; in final form: 11 October 2022; accepted: 12 October 2022; published online: 24 October 2022
DOI: 10.12921/cmst.2022.0000022
Abstract:
We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative one-dimensional heat-conducting oscillator exposed to a temperature gradient in a three-dimensional phase space with coordinate q, momentum p, and thermostat control variable ζ. The second type simulates a conservative two-dimensional N -body fluid with 4N phase variables {q, p} undergoing shock compression. Despite the time-reversibility of each of the three oscillator equations and all of the 4N manybody motion equations both types of simulation are irreversible, obeying the Second Law of Thermodynamics. But for different reasons. The irreversible oscillator seeks out an attractive dissipative limit cycle. The likewise irreversible, but thoroughly conservative, Newtonian shockwave eventually generates a reversible near-equilibrium pair of rarefaction fans. Both problem types illustrate interesting features of Lyapunov instability. This instability results in the exponential growth of small perturbations, ∝ e^(λt) where λ is a “Lyapunov exponent”.
Key words:
heat conduction, irreversibility, shockwaves, thermodynamics’ Second Law, time reversibility
References:
[1] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Method, The Journal of Chemical Physics 81, 511–519 (1984).
[2] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[3] Wm.G. Hoover, Canonical Dynamics. Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[4] 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).
[5] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative Attractors in Three and Four Phase-Space Dimensions, Physical Review E 55, 6803–6810 (1997).
[6] 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).
[7] 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).
[8] 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).
[9] W.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).
[10] W.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics. Part III. A Memoir for Douglas James Henderson [1934–2020], Computational Methods in Science and Technology 26, 111–120 (2020).
[11] J.C. Sprott, W.G. Hoover, C.G. Hoover, Elegant Simulations, World Scientific, Singapore (2023).
[12] M. Ross, B.J. Alder, Shock Compression of Argon II. Nonadditive Repulsive Potential, The Journal of Chemical Physics 46, 4203–4210 (1967).
We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative one-dimensional heat-conducting oscillator exposed to a temperature gradient in a three-dimensional phase space with coordinate q, momentum p, and thermostat control variable ζ. The second type simulates a conservative two-dimensional N -body fluid with 4N phase variables {q, p} undergoing shock compression. Despite the time-reversibility of each of the three oscillator equations and all of the 4N manybody motion equations both types of simulation are irreversible, obeying the Second Law of Thermodynamics. But for different reasons. The irreversible oscillator seeks out an attractive dissipative limit cycle. The likewise irreversible, but thoroughly conservative, Newtonian shockwave eventually generates a reversible near-equilibrium pair of rarefaction fans. Both problem types illustrate interesting features of Lyapunov instability. This instability results in the exponential growth of small perturbations, ∝ e^(λt) where λ is a “Lyapunov exponent”.
Key words:
heat conduction, irreversibility, shockwaves, thermodynamics’ Second Law, time reversibility
References:
[1] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Method, The Journal of Chemical Physics 81, 511–519 (1984).
[2] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[3] Wm.G. Hoover, Canonical Dynamics. Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[4] 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).
[5] H.A. Posch, W.G. Hoover, Time-Reversible Dissipative Attractors in Three and Four Phase-Space Dimensions, Physical Review E 55, 6803–6810 (1997).
[6] 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).
[7] 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).
[8] 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).
[9] W.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).
[10] W.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics. Part III. A Memoir for Douglas James Henderson [1934–2020], Computational Methods in Science and Technology 26, 111–120 (2020).
[11] J.C. Sprott, W.G. Hoover, C.G. Hoover, Elegant Simulations, World Scientific, Singapore (2023).
[12] M. Ross, B.J. Alder, Shock Compression of Argon II. Nonadditive Repulsive Potential, The Journal of Chemical Physics 46, 4203–4210 (1967).
