Abstract
We consider simulations of a two-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing Lyapunov exponents. To this spectrum are associated “eigen-directions”, called Lyapunov modes. We carefully analyze these modes and show how they are naturally associated with vector fields over the container. We also show that the Lyapunov exponents, and the coupled dynamics of the modes (where it exists) follow linear laws, whose coefficients only depend on the density of the gas, but not on aspect ratio and very little on the boundary conditions.
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References
Lj Milanović HA Posch Wm.G. Hoover (1998) ArticleTitleWhat is ‘Liquid’? Understanding the states of matter Mol. Phys. 95 281–287
H. A. Posch and R. Hirschl, Simulation of Billiards and of Hard-Body Fluids, in Hard Ball Systems and the Lorenz Gas, D. Szász, ed. Encyclopedia of the mathematical sciences Vol. 101, (Springer Verlag, Berlin, 2000).
J.-P. Eckmann O. Gat (2000) ArticleTitleHydrodynamic Lyapunov modes in translation-invariant systems J. Stat. Phys. 98 775
S. McNamara M. Mareschal (2001) ArticleTitleThe Lyapunov spectrum of granular gases Phys. Rev. E. 63 061306
S. McNamara M. Mareschal (2001) ArticleTitleOn the origin of the hydrodynamic Lyapunov modes Phys. Rev. E. 64 051103
T. Taniguchi G. P. Morriss (2003) ArticleTitleBoundary effects in the stepwise structure of the Lyapunov spectra for quasi-one-dimensional systems Phys. Rev. E. 68 026218
A. Wijn Particlede H. Beijeren Particlevan (2004) ArticleTitleGoldstone modes in Lyapunov spectra of hard sphere systems Phys. Rev. E. 70 016207
Ch Foster R Hirschl HA Posch Wm.G. Hoover (2004) ArticleTitlePerturbed phase-space dynamics of hard-disk fluids Physica D. 187 294
V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical Systems, Trudy Mosk. Mat. Obsc . 19 :179 [ Moscow Math. Soc . 19 :197 (1968)].
JP Eckmann D Ruelle (1985) ArticleTitleErgodic theory of chaos and strange attractors Rev. Mod. Phys. 57 617
Lj. Milanović HA Posch (2002) ArticleTitleLocalized and delocalized modes in the tangent-space dynamics of planar hard dumbbell fluids J. Mol. Liquids. 96–97 221
Ch. Dellago H. A. Posch Wm. G. Hoover (1996) ArticleTitleLyapunov instability of hard disks in equilibrium nonequilibrium steady states Phys. Rev. E. 53 1485
G. Benettin L. Galgani A. Giorgilli J.M. Strelcyn (1980) Meccanica 15 21
I. Shimada T. Nagashima (1979) ArticleTitleA numerical approach to ergodic problem of dissipative dynamical system Prog. Theor. Phys. 61 1605
N Chernov (1983) ArticleTitleConstruction of transverse fiberings in multi-dimensional semidispersed billiards Funct. Anal. Appl. 16 270–280
M. Wojtkowski, Systems of classical interacting particles with non-vanishing Lyapunov exponents in Lyapunov exponents (Oberwolfach, 1990), Lecture Notes in Math. Vol. 1486 (Springer, 1991).
L. Bunimovich C. Liverani A. Pellegrinotti Y. Suhov (1992) ArticleTitleErgodic systems of n balls in a billiard table Commun. Math. Phys. 146 357
N Simányi D Szász (1999) ArticleTitleHard ball systems are completely hyperbolic Ann. Math. 149 35–96
N Simányi (2002) ArticleTitleThe complete hyperbolicity of cylindric billiard Erg. Th. Dyn. Syst. 22 281–302
N Simányi (2002) ArticleTitleProof of the ergodic hypothesis for typical hard ball systems Ann. Henri Poincaré 5 203–233
M Wojtkowski (1988) ArticleTitleMeasure theoretic entropy of the system of hard spheres Ergod. Th . & Dynam. 8 133
Ch. Dellago HA Posch (1997) ArticleTitleKolmogorov–Sinai entropy and Lyapunov spectra of a hard sphere gas Physica D. 240 68
Ch. Forster D. Mukamel H. A. Posch (2004) ArticleTitleHard disk in narrow channels Phys. Rev. E. 69 066124
Ch. Forster, R. Hirschl, and H. A. Posch, Analysis of Lyapunov modes for hard-disk systems, Proceedings of ICMP 2003.
R. Hirschl, Computer simulation of hard-disk systems: ‘‘Lyapunov modes’’ and stochastic color conductivity, diploma thesis, University of Vienna (1999).
Ch. Forster and H. A. Posch, Lyapunov modes for soft-disk fluids, New J. Phys. 7 , article 32 (2005).
G. Radons and H. Yang, Static and dynamic correlations in many-particle Lyapunov vectors, submitted to Phys. Rev. Lett. (2004).
H. Yang and G. Radons, Lyapunov instabilities of Lennard-Jones fluids, Phys. Rev. E , submitted (2004).
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Eckmann, JP., Forster, C., Posch, H.A. et al. Lyapunov Modes in Hard-Disk Systems. J Stat Phys 118, 813–847 (2005). https://doi.org/10.1007/s10955-004-2687-4
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DOI: https://doi.org/10.1007/s10955-004-2687-4