Time - Symmetry Breaking in Hamiltonian Mechanics

• Autorzy: Hoover W. G. , Hoover C. G.
• Języki publikacji: EN

Abstrakty

EN

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: {+p }→{-g}. Here we adopt Levesque and Verlet’s precisely bit-reversible motion algorithm to ensure that the trajectory reversibility is exact, with the forward and backward sets of coordinates identical. Nevertheless, the associated instantaneous Lyapunov instability, or “sensitive dependence on initial conditions” of “chaotic” (or “Lyapunov unstable”) bit-reversible coordinate trajectories can still exhibit an exponentially growing time-symmetry-breaking irreversibility ≃ eλt. Surprisingly, the positive and negative exponents, as well as the forward and backward Lyapunov spectra , {λforward(t) } and {λt backward(t) }, are usually not closely related, and so give four differing topological measures of “local” chaos. We have demonstrated this symmetry breaking for fluid shockwaves, for free expansions, and for chaotic molecular collisions. Here we illustrate and discuss this time-symmetry breaking for three statistical-mechanical systems, [i] a minimal (but still chaotic) one-body “cell model” with a four-dimensional phase space; [ii] relatively small colliding crystallites, for which the whole Lyapunov spectrum is accessible; [iii] a near-continuum inelastic collision of two larger 400-particle balls. In the last two of these pedagogical problems the two colliding bodies coalesce. The particles most prone to Lyapunov instability are dramatically different in the two time directions. Thus this Lyapunov-based symmetry breaking furnishes an interesting Arrow of Time.

Słowa kluczowe

EN

reversibility; Lyapunov instability; inelastic collisions; time-symmetry breaking

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2013
• Tom: Vol. 19, No. 2
• Strony: 77--87
• Opis fizyczny: Bibliogr. 15 poz., rys.

Twórcy

autor

Hoover W. G.

• Ruby Valley Research Institute Highway Contract 60, Box 601 Ruby Valley, Nevada 89833

autor

Hoover C. G.

• Ruby Valley Research Institute Highway Contract 60, Box 601 Ruby Valley, Nevada 89833

Bibliografia

• [1] Wm.G. Hoover and C.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 C.G. Hoover, Time’s Arrow for Shockwaves; Bit-Reversible Lyapunov and Covariant Vectors ; Symmetry Breaking, Computational Methods in Science and Technology 19(2), 69-75 (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 C.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).