Time’s Arrow for Shockwaves ; Bit-Reversible Lyapunov and “Covariant” Vectors ; Symmetry Breaking
Ruby Valley Research Institute, Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
(Received: 29 January 2013; accepted: 12 February 2013; published online: 8 March 2013)
DOI: 10.12921/cmst.2013.19.02.69-75
OAI: oai:lib.psnc.pl:429
Abstract:
Strong shockwaves generate entropy quickly and locally. The Newton-Hamilton equations of motion, which underly the dynamics, are perfectly time-reversible. How do they generate the irreversible shock entropy? What are the symptoms of this irreversibility? We investigate these questions using Levesque and Verlet’s bit-reversible algorithm. In this way we can generate an entirely imaginary past consistent with the irreversibility observed in the present. We use Runge-Kutta integration to analyze the local Lyapunov instability of nearby “satellite” trajectories. From the forward and backward processes we identify those particles most intimately connected with the irreversibility described by the Second Law of Thermodynamics. Despite the perfect time symmetry of the particle trajectories, the fully-converged vectors associated with the largest Lyapunov exponents, forward and backward in time, are qualitatively different. The vectors display a time-symmetry breaking equivalent to Time’s Arrow. That is, in autonomous Hamiltonian shockwaves the largest local Lyapunov exponents, forward and backward in time, are quite different.
Key words:
bit reversibility, Lyapunov instability, shockwaves, time reversibility
References:
[1] B. L. Holian, Wm. G. Hoover, and H. A. Posch, “Resolution of Loschmidt’s Paradox: the Origin of Irreversible Behavior in Reversible Atomistic Dynamics”, Physical Review Letters 59, 10-13 (1987).
[2] Wm. G. Hoover and Carol G. Hoover, Time Reversibility, Computer Simulation, Algorithms, and Chaos (World Scientific, Singapore, 2012).
[3] Wm. G. Hoover, “Liouville’s Theorems, Gibbs’ Entropy, and Multifractal Distributions for Nonequilibrium Steady States”, Journal of Chemical Physics 109, 4164-4170 (1998).
[4] D. Levesque and L. Verlet, “Molecular Dynamics and Time Reversibility”, Journal of Statistical Physics 72, 519-537 (1993).
[5] M. Romero-Bastida, D. Pazó, J M. López, and M. A. Rodríguez, “Structure of Characteristic Lyapunov Vectors in Anharmonic Hamiltonian Lattices”, Physical Review E 82, 036205 (2010).
[6] Wm. G. Hoover and Carol G. Hoover, “Three Lectures; NEMD, SPAM, and Shockwaves”, pages 23-55 in Nonequilibrium Statistical Physics Today (Proceedings of the 11th Granada Seminar on Computational and Statistical Physics, 13-17 September 2010), P. L. Garrido, J. Marro, and F. de los Santos, Editors (AIP Conference Proceedings #1332, Melville, New York, 2011).
[7] Wm. G. Hoover and H. A. Posch, “Direct Measurement of Equilibrium and Nonequilibrium Lyapunov Spectra”, Physics Letters A 123, 227-230 (1987).
[8] Wm. G. Hoover, C. G. Hoover, and H. A. Posch, “Dynamical Instabilities, Manifolds, and Local Lyapunov Spectra Far From Equilibrium”, Computational Methods in Science and Technology (Poznan, Poland) 7, 55-65 (2001).
[9] P. V. Kuptsov and U. Parlitz, “Theory and Computation of Covariant Lyapunov Vectors”, Journal of Nonlinear Science 22, 727-762 (2012); ariv 1105.5228v3.
[10] C. Dellago and Wm. G. Hoover, “Are Local Lyapunov Exponents Continuous in Phase Space?”, Physics Letters A 268, 330-334 (2000).
[11] Wm. G. Hoover, C. G. Hoover, and D. J. Isbister, “Chaos, Ergodic Convergence, and Fractal Instability for a Thermostated Canonical Harmonic Oscillator, Physical Review E 63, 026209 (2001).
Strong shockwaves generate entropy quickly and locally. The Newton-Hamilton equations of motion, which underly the dynamics, are perfectly time-reversible. How do they generate the irreversible shock entropy? What are the symptoms of this irreversibility? We investigate these questions using Levesque and Verlet’s bit-reversible algorithm. In this way we can generate an entirely imaginary past consistent with the irreversibility observed in the present. We use Runge-Kutta integration to analyze the local Lyapunov instability of nearby “satellite” trajectories. From the forward and backward processes we identify those particles most intimately connected with the irreversibility described by the Second Law of Thermodynamics. Despite the perfect time symmetry of the particle trajectories, the fully-converged vectors associated with the largest Lyapunov exponents, forward and backward in time, are qualitatively different. The vectors display a time-symmetry breaking equivalent to Time’s Arrow. That is, in autonomous Hamiltonian shockwaves the largest local Lyapunov exponents, forward and backward in time, are quite different.
Key words:
bit reversibility, Lyapunov instability, shockwaves, time reversibility
References:
[1] B. L. Holian, Wm. G. Hoover, and H. A. Posch, “Resolution of Loschmidt’s Paradox: the Origin of Irreversible Behavior in Reversible Atomistic Dynamics”, Physical Review Letters 59, 10-13 (1987).
[2] Wm. G. Hoover and Carol G. Hoover, Time Reversibility, Computer Simulation, Algorithms, and Chaos (World Scientific, Singapore, 2012).
[3] Wm. G. Hoover, “Liouville’s Theorems, Gibbs’ Entropy, and Multifractal Distributions for Nonequilibrium Steady States”, Journal of Chemical Physics 109, 4164-4170 (1998).
[4] D. Levesque and L. Verlet, “Molecular Dynamics and Time Reversibility”, Journal of Statistical Physics 72, 519-537 (1993).
[5] M. Romero-Bastida, D. Pazó, J M. López, and M. A. Rodríguez, “Structure of Characteristic Lyapunov Vectors in Anharmonic Hamiltonian Lattices”, Physical Review E 82, 036205 (2010).
[6] Wm. G. Hoover and Carol G. Hoover, “Three Lectures; NEMD, SPAM, and Shockwaves”, pages 23-55 in Nonequilibrium Statistical Physics Today (Proceedings of the 11th Granada Seminar on Computational and Statistical Physics, 13-17 September 2010), P. L. Garrido, J. Marro, and F. de los Santos, Editors (AIP Conference Proceedings #1332, Melville, New York, 2011).
[7] Wm. G. Hoover and H. A. Posch, “Direct Measurement of Equilibrium and Nonequilibrium Lyapunov Spectra”, Physics Letters A 123, 227-230 (1987).
[8] Wm. G. Hoover, C. G. Hoover, and H. A. Posch, “Dynamical Instabilities, Manifolds, and Local Lyapunov Spectra Far From Equilibrium”, Computational Methods in Science and Technology (Poznan, Poland) 7, 55-65 (2001).
[9] P. V. Kuptsov and U. Parlitz, “Theory and Computation of Covariant Lyapunov Vectors”, Journal of Nonlinear Science 22, 727-762 (2012); ariv 1105.5228v3.
[10] C. Dellago and Wm. G. Hoover, “Are Local Lyapunov Exponents Continuous in Phase Space?”, Physics Letters A 268, 330-334 (2000).
[11] Wm. G. Hoover, C. G. Hoover, and D. J. Isbister, “Chaos, Ergodic Convergence, and Fractal Instability for a Thermostated Canonical Harmonic Oscillator, Physical Review E 63, 026209 (2001).
