Hydrodynamic Lyapunov modes, which have recently been observed in many extended systems with translational symmetry, such as hard sphere systems, dynamic XY models or Lennard–Jones fluids, are nowadays regarded as fundamental objects connecting nonlinear dynamics and statistical physics. We review here our recent results on Lyapunov modes in extended system. The solution to one of the puzzles, the appearance of good and ‘vague’ modes, is presented for the model system of coupled map lattices. The structural properties of these modes are related to the phase space geometry, especially the angles between Oseledec subspaces, and to fluctuations of local Lyapunov exponents. In this context, we report also on the possible appearance of branches splitting in the Lyapunov spectra of diatomic systems, similar to acoustic and optical branches for phonons. The final part is devoted to the hyperbolicity of partial differential equations and the effective degrees of freedom of such infinite-dimensional systems.

References

Benettin G., Galgani L.& Strelcyn J. M.. 1976Kolmogorov entropy and numerical experiments. Phys. Rev. A 14, 2338–2345. (doi:10.1103/PhysRevA.14.2338). Crossref, Google Scholar
Bochi J.& Viana M.. 2005The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. Math. 161, 1423–1485. (doi:10.4007/annals.2005.161.1423). Crossref, Google Scholar
Cross M. C.& Hohenberg P. C.. 1993Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112. (doi:10.1103/RevModPhys.65.851). Crossref, ISI, Google Scholar
de Wijn A. S.& van Beijeren H.. 2004Goldstone modes in Lyapunov spectra of hard sphere systems. Phys. Rev. E 70, 016 207. (doi:10.1103/PhysRevE.70.016207). Crossref, Google Scholar
Dorfman J. P.. 1999An introduction to chaos in nonequilibrium statistical mechanicsCambridge, UK: Cambridge University Press. Crossref, Google Scholar
Eckmann J.-P., Forster C., Posch H. A.& Zabey E.. 2005Lyapunov modes in hard-disk systems. J. Stat. Phys. 118, 813–847. (doi:10.1007/s10955-004-2687-4). Crossref, Google Scholar
Foias C., Sell G. R.& Témam R.. 1988Inertial manifolds for nonlinear evolutionary equations. J. Differ. Equ. 73, 309–353. (doi:10.1016/0022-0396(88)90110-6). Crossref, ISI, Google Scholar
Forster C.& Posch H. A.. 2005Lyapunov modes in soft-disk fluids. New J. Phys. 7, 1367–2630. (doi:10.1088/1367-2630/7/1/032). Crossref, Google Scholar
Forster C., Hirschl R., Posch H. A.& Hoover W. G.. 2004Perturbed phase-space dynamics of hard-disk fluids. Phys. D 187, 294–310. (doi:10.1016/j.physd.2003.09.013). Crossref, Google Scholar
Ginelli F., Poggi P., Turchi A., Chaté H., Livi R.& Politi A.. 2007Characterizing dynamics with covariant Lyapunov vectors. Phys. Rev. Lett. 99, 130 601. (doi:10.1103/PhysRevLett.99.130601). Crossref, Google Scholar
Hoover W. G., Posch H. A., Forster C., Dellago C.& Zhou M.. 2002Lyapunov modes of two-dimensional many-body systems; soft disks, hard disks, and rotors. J. Stat. Phys. 109, 765–775. (doi:10.1023/A:1020474901341). Crossref, Google Scholar
Keyes T.. 1997Instantaneous normal mode approach to liquid state dynamics. J. Phys. Chem. A 101, 2921–2930. (doi:10.1021/jp963706h). Crossref, Google Scholar
Krylov N. S.. 1979Works on the foundations of statistical mechanicsPrinceton, NJ: Princeton University Press. Google Scholar
Pettini M., Casetti L., Cerruti-Sola M., Franzosi R.& Cohen E. D.. 2005Weak and strong chaos in Fermi—Pasta—Ulam models and beyond. Chaos 15, 015 106. (doi:10.1063/1.1849131). Crossref, Google Scholar
Posch H. A.& Hirschl R.. 2000Simulation of billiards and of hard body fluids. Hard ball systems and the Lorentz gas (ed. & Szàsz D.). Berlin, Germany: Springer. Google Scholar
Radons G.& Yang H. L.. 2004Static and dynamic correlations in many-particle Lyapunov vectors. (http://arxiv.org/abs/nlin/0404028). Google Scholar
Ruelle D.. 1982Large volume limit of the distribution of characteristic exponents in turbulence. Commun. Math. Phys. 87, 287–302. (doi:10.1007/BF01218566). Crossref, Google Scholar
Sinai Ya. G.. 2000Dynamical systems, ergodic theory and applicationsBerlin, Germany: Springer. Google Scholar
Taniguchi T.& Morriss G. P.. 2003Boundary effects in the stepwise structure of the Lyapunov spectra for quasi-one-dimensional systems. Phys. Rev. E 68, 026 218. (doi:10.1103/PhysRevE.68.026218). Crossref, Google Scholar
Taniguchi T.& Morriss G. P.. 2005Time-oscillating Lyapunov modes and the momentum autocorrelation function. Phys. Rev. Lett. 94, 154 101. (doi:10.1103/PhysRevLett.94.154101). Crossref, Google Scholar
Yang H. L.& Radons G.. 2005Lyapunov instability of Lennard—Jones fluids. Phys. Rev. E 71, 036 211. (doi:10.1103/PhysRevE.71.036211). Crossref, Google Scholar
Yang H. L.& Radons G.. 2006aUniversal features of hydrodynamic Lyapunov modes in extended systems with continuous symmetries. Phys. Rev. Lett. 96, 074 101. (doi:10.1103/PhysRevLett.96.074101). Crossref, Google Scholar
Yang H. L.& Radons G.. 2006bHydrodynamic Lyapunov modes in coupled map lattices. Phys. Rev. E 73, 016 202. (doi:10.1103/PhysRevE.73.016202). Crossref, Google Scholar
Yang H. L.& Radons G.. 2006cDynamical behavior of hydrodynamic Lyapunov modes in coupled map lattices. Phys. Rev. E 73, 016 208. (doi:10.1103/PhysRevE.73.016208). Crossref, Google Scholar
Yang H. L.& Radons G.. 2006dHydrodynamic Lyapunov modes and strong stochasticity threshold in Fermi—Pasta—Ulam models. Phys. Rev. E 73, 066 201. (doi:10.1103/PhysRevE.73.066201). Crossref, Google Scholar
Yang H. L.& Radons G.. 2007Lyapunov spectral gap and branch splitting of Lyapunov modes in a diatomic system. Phys. Rev. Lett. 99, 164 101. (doi:10.1103/PhysRevLett.99.164101). Crossref, Google Scholar
Yang H. L.& Radons G.. 2008aWhen can one observe good hydrodynamic Lyapunov modes?. Phys. Rev. Lett. 100, 024 101. (doi:10.1103/PhysRevLett.100.024101). Crossref, Google Scholar
Yang H. L.& Radons G.. 2008bHydrodynamic Lyapunov modes and strong stochasticity threshold in the dynamic XY model: an alternative scenario. Phys. Rev. E 77, 016 203. (doi:10.1103/PhysRevE.77.016203). Crossref, Google Scholar
Yang H. L.& Radons G.. 2008cLyapunov instabilities of extended systems. NIC series, vol. 39 (eds , Münster G., Wolf D.& Kremer M.), pp. 349–358. Google Scholar
Yang H. L., Takeuchi K. A., Ginelli F., Chaté H.& Radons G.. 2009Hyperbolicity and the effective dimension of spatially-extended dissipative systems. Phys. Rev. Lett. 102, 074 102. (doi:10.1103/PhysRevLett.102.074102). Crossref, Google Scholar

This Issue

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

28 August 2009

Volume 367Issue 1901

Theme Issue 'Topics on non-equilibrium statistical mechanics and nonlinear physics' compiled by Orazio Descalzi, Arturo C. Martí, Cristina Masoller and Osvaldo A. Rosso

Article Information

DOI:https://doi.org/10.1098/rsta.2009.0067
Published by:Royal Society
Print ISSN:1364-503X
Online ISSN:1471-2962

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Published online28/08/2009
Published in print28/08/2009

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statistical physics