Abstract
The tangent dynamics of the Lyapunov modes and their dynamics as generated numerically—the numerical dynamics—is considered. We present a new phenomenological description of the numerical dynamical structure that accurately reproduces the experimental data for the quasi-one-dimensional hard-disk system, and shows that the Lyapunov mode numerical dynamics is linear and separate from the rest of the tangent space. Moreover, we propose a new, detailed structure for the Lyapunov mode tangent dynamics, which implies that the Lyapunov modes have well-defined (in)stability in either direction of time. We test this tangent dynamics and its derivative properties numerically with partial success. The phenomenological description involves a time-modal linear combination of all other Lyapunov modes on the same polarization branch and our proposed Lyapunov mode tangent dynamics is based upon the form of the tangent dynamics for the zero modes.
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References
Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)
Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method of computing all of them. Meccanica 15, 9 (1980)
de Wijn, A., van Biejeren, H.: Goldstone modes in Lyapunov spectra of hard sphere systems. Phys. Rev. E 70, 016, 207 (2004)
Dellago, C., Posch, H., Hoover, W.: Lyapunov instability in a system of hard disks in equilibrium and nonequilibrium steady states. Phys. Rev. E 53, 1485 (1996)
Eckmann, J.P., Forster, C., Posch, H., Zabey, E.: Lyapunov modes in hard-disk systems. J. Stat. Phys. 118, 813 (2005)
Eckmann, J.P., Gat, O.: Hydrodynamic Lyapunov modes in translation-invariant systems. J. Stat. Phys. 98, 775 (2000)
Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617 (1985)
Ershov, S.V., Potapov, A.B.: On the concept of stationary Lyapunov basis. Physica D 118, 167 (1998)
Forster, C., Hirschl, R., Posch, H.A., Hoover, W.G.: Perturbed phase-space dynamics of hard sphere systems. Physica D 187, 294 (2004)
McNamara, S., Mareschal, M.: Origin of the hydrodynamic Lyapunov modes. Phys. Rev. E 64, 051, 103 (2001)
Milanovic, L., Posch, H.A.: Localized and delocalized modes in the tangent-space dynamics of planar hard dumbbell fluids. J. Mol. Liq. 96–97, 221 (2002)
Milanovic, L., Posch, H.A., Hoover, W.: What is ‘liquid’? understanding the states of matter. Mol. Phys. 95, 281 (1998)
Oseledec, V.I.: A multiplicative ergodic theorem Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197 (1968)
Pikovsky, A., Politi, A.: Dynamic localization of Lyapunov exponents in spacetime chaos. Nonlinearity 11, 1049 (1998)
Posch, H.A., Hirschl, R.: Hard-Ball Systems and the Lorentz Gas. Springer, Berlin (2000)
Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50, 27 (1979)
Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical system. Prog. Theor. Phys. 61, 1605 (1979)
Taniguchi, T., Morriss, G.P.: Stepwise structure of Lyapunov spectra for many-particle systems using a random matrix dynamics. Phys. Rev. E 65, 056, 202 (2002)
Taniguchi, T., Morriss, G.P.: Time-dependent mode structure for Lyapunov vectors as a collective movement in quasi-one-dimensional systems. Phys. Rev. E 71, 016, 218 (2005)
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Robinson, D.J., Morriss, G.P. Lyapunov Mode Dynamics in Hard-Disk Systems. J Stat Phys 131, 1–31 (2008). https://doi.org/10.1007/s10955-007-9473-z
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DOI: https://doi.org/10.1007/s10955-007-9473-z