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Volume 7 (1) 2001, 55-65


Hoover Wm.G. , Hoover C.G. 1,2, Posch H. A. 3

1 Department ofApplied Science, University of California at Davis,
Lawrence Livermore National Laboratory, Livermore 94550, California, USA
2 Methods Development Group, Department of Mechanical Engineering,
Lawrence Livermore National Laboratory, Livermore 94550, California, USA
3 Institut für Experimentalphysik, Universität Wien,
Boltzmanngasse 5, A-1090 Wien, Austria

DOI:   10.12921/cmst.2001.07.01.55-65



We consider an harmonic oscillator in a thermal gradient far from equilibrium. The motion is made ergodic and fully time-reversible through the use of two thermostat variables. The equations of motion contain both linear and quadratic terms. The dynamics is chaotic. The resulting phase-space distribution is not only complex and multifiactal, but also ergodic, due to the time-reversibility property. We analyze dynamical time series in two ways. We describe local, but comoving, singularities in terms of the “local Lyapunov spectrum” {λ}. Local singularities at a particular phase-space point can alternatively be described by the local eigenvalues and eigenvectors of the “dynamical matrix” D is the matrix of derivatives of the equations of motion We pursue this eigenvalue-eigenvector description for the oscillator. We find that it breaks down at a dense set of singular points, where the four eigenvectors span only a three-dimensional subspace. We believe that the concepts of stable and unstable global manifolds are problematic for this simple nonequilibrium system.


[1] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.
[2] Wm. G. Hoover, Time Reversibility, Computer Simulation, and Chaos, World Scientific, Singapore, 1999.
[3] Wm. G. Hoover, Computational Statistical Mechanics, Elsevier, New York, 1991.
[4] H. A. Posch and Wm. G. Hoover, Phys. Rev., E55, 6803 (1997).
[5] Ch. Dellago and Wm. G. Hoover, Phys. Lett., A268, 330 (2000).
[6] Wm. G. Hoover, C. G. Hoover, and D. J. Isbister, Chaos, Ergodic Convergence, and Fractal
Instability for a Thermostated Canonical Harmonic Oscillator, Phys. Rev. E 63, 026209 (2000).
[7] P. Gaspard, Chaos, Scattering, and Statistical Mechanics, Cambridge University Press, 1998. It is a pleasure to thank Rainer Klages and Pierre Gaspard for their interesting comments on this point.
[8] See both the articles and the discussion remarks in the two useful summary volumes: Microscopic Simulations of Complex Flows, ed. M. Mareschal, Plenum, New York, 1989 and Microscopic Simulations of Complex Hydrodynamic Phenomena, ed. M. Mareschal and B. L. Holian, Plenum, New York, 1992.
[9] D. J. Evans, E. G. D. Cohen, G. P. Morriss, Phys. Rev. Lett., 71, 2401 (1993). We are indebted to Denis Evans for this estimate of the relaxation time.
[10] G. 1. Martyna, M. L. Klein, and M. E. Tuckerman, J. Chem. Phys., 97, 2635 (1992).
[11] Numerical methods were first developed by Giancarlo Benettin’s group, following related work by Spotswood Stoddard and Joseph Ford.
[12] D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic, New York, 1990.