Time-Symmetry Breaking in Hamiltonian Mechanics
Ruby Valley Research Institute, Highway Contract 60, Box 601
Ruby Valley, Nevada 89833
E-mail: hooverwilliam@yahoo.com
Received:
(Received: 11 February 2013; revised: 8 March 2013; accepted: 11 March 2013; published online: 13 March 2013)
DOI: 10.12921/cmst.2013.19.02.77-87
OAI: oai:lib.psnc.pl:432
Abstract:
Hamiltonian trajectories are strictly time-reversible. Any time series of Hamiltonian coordinates f q g satisfying Hamilton’s motion equations will likewise satisfy them when played “backwards”, with the corresponding momenta changing signs : f +p g
Key words:
Inelastic Collisions, Lyapunov instability, reversibility, time-symmetry breaking
References:
[1] Wm. G. Hoover and Carol G. Hoover, Time Reversibility, Computer Simulation, Algorithms, and Chaos (World Scientific, Singapore, 2012).
[2] Wm. G. Hoover, Computational Statistical Mechanics (Elsevier Science, 1991), available free of charge at our website
[ www.williamhoover.info ].
[3] S. D. Stoddard and J. Ford, “Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System”, Physical Review A 8, 1504-1512 (1973).
[4] G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; a Method for Computing All of Them”, Meccanica 15, 9-30 (1980).
[5] W. G. Hoover and H. A. Posch, “Direct Measurement of Lyapunov Exponents”, Physics Letters A 113, 82-84 (1985),
[6] H. A. Posch, Wm. G. Hoover, and F. J. Vesely, “Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos”, Physical Review A 33, 4253-4265 (1986).
[7] Wm. G. Hoover, C. G. Hoover, I. F. Stowers, A. J. De Groot, and B. Moran, “Simulation of Mechanical Deformation via Nonequilibrium Molecular Dynamics”, in Microscopic Simulations of Complex Flows, Edited by Michel Mareschal (Volume 236 of NATO Science Series B, Plenum Press, 1990).
[8] 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).
[9] Wm. G. Hoover and Carol G. Hoover, “Time’s Arrow for Shockwaves; Bit-Reversible Lyapunov and Covariant Vectors ; Symmetry Breaking”, Computational Methods in Science and Technology 19(2), 5-11 (2013).
[10] M. Romero-Bastida, D. Pazó, J. M. Lopéz, and M. A. Rodríguez, “Structure of Characteristic Lyapunov Vectors in Anharmonic Hamiltonian Lattices”, Physical Review E 82, 036205 (2010).
[11] D. Levesque and L. Verlet, “Molecular Dynamics and Time Reversibility”, Journal of Statistical Physics 72, 519-537 (1993).
[12] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Incorporated (New York, 1954).
[13] S. M. Foiles, M. I. Baskes, and M. S. Daw, “Embedded-Atom- Method Functions for the FCC Metals Cu, Ag, Au, Ni, Pd, Pt, and their Alloys”, Physical Review B 33, 7983-7991 (1986).
[14] J. L. Lebowitz, “Boltzmann’s Entropy and Time’s Arrow”, Physics Today 46, 32-38 (September, 1993).
[15] F. J. Uribe, Wm. G. Hoover and Carol G. Hoover, “Maxwell and Cattaneo’s Time-Delay Ideas Applied to Shockwaves and the Rayleigh-Bénard Problem”, Computational Methods in Science and Technology 19(1), 5-12 (online January 2013).
Hamiltonian trajectories are strictly time-reversible. Any time series of Hamiltonian coordinates f q g satisfying Hamilton’s motion equations will likewise satisfy them when played “backwards”, with the corresponding momenta changing signs : f +p g
Key words:
Inelastic Collisions, Lyapunov instability, reversibility, time-symmetry breaking
References:
[1] Wm. G. Hoover and Carol G. Hoover, Time Reversibility, Computer Simulation, Algorithms, and Chaos (World Scientific, Singapore, 2012).
[2] Wm. G. Hoover, Computational Statistical Mechanics (Elsevier Science, 1991), available free of charge at our website
[ www.williamhoover.info ].
[3] S. D. Stoddard and J. Ford, “Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System”, Physical Review A 8, 1504-1512 (1973).
[4] G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, “Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; a Method for Computing All of Them”, Meccanica 15, 9-30 (1980).
[5] W. G. Hoover and H. A. Posch, “Direct Measurement of Lyapunov Exponents”, Physics Letters A 113, 82-84 (1985),
[6] H. A. Posch, Wm. G. Hoover, and F. J. Vesely, “Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos”, Physical Review A 33, 4253-4265 (1986).
[7] Wm. G. Hoover, C. G. Hoover, I. F. Stowers, A. J. De Groot, and B. Moran, “Simulation of Mechanical Deformation via Nonequilibrium Molecular Dynamics”, in Microscopic Simulations of Complex Flows, Edited by Michel Mareschal (Volume 236 of NATO Science Series B, Plenum Press, 1990).
[8] 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).
[9] Wm. G. Hoover and Carol G. Hoover, “Time’s Arrow for Shockwaves; Bit-Reversible Lyapunov and Covariant Vectors ; Symmetry Breaking”, Computational Methods in Science and Technology 19(2), 5-11 (2013).
[10] M. Romero-Bastida, D. Pazó, J. M. Lopéz, and M. A. Rodríguez, “Structure of Characteristic Lyapunov Vectors in Anharmonic Hamiltonian Lattices”, Physical Review E 82, 036205 (2010).
[11] D. Levesque and L. Verlet, “Molecular Dynamics and Time Reversibility”, Journal of Statistical Physics 72, 519-537 (1993).
[12] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Incorporated (New York, 1954).
[13] S. M. Foiles, M. I. Baskes, and M. S. Daw, “Embedded-Atom- Method Functions for the FCC Metals Cu, Ag, Au, Ni, Pd, Pt, and their Alloys”, Physical Review B 33, 7983-7991 (1986).
[14] J. L. Lebowitz, “Boltzmann’s Entropy and Time’s Arrow”, Physics Today 46, 32-38 (September, 1993).
[15] F. J. Uribe, Wm. G. Hoover and Carol G. Hoover, “Maxwell and Cattaneo’s Time-Delay Ideas Applied to Shockwaves and the Rayleigh-Bénard Problem”, Computational Methods in Science and Technology 19(1), 5-12 (online January 2013).
