Perturbed phase-space dynamics of hard-disk fluids
  The Lyapunov spectrum describes the exponential growth, or decay, of infinitesimal phase-space perturbations. The perturbation associated with the maximum Lyapunov exponent is strongly localized in space, and only a small fraction of all particles contributes to the perturbation growth at any instant of time. This fraction converges to zero in the thermodynamic large-particle-number limit. For hard-disk and hard-sphere systems the perturbations belonging to the small positive and large negative exponents are coherently spread out and form orthogonal periodic structures in space, the ``Lyapunov modes''. There are two types of mode polarizations, transverse and longitudinal. The transverse modes do not propagate, but the longitudinal modes do with a speed about one third of the sound speed. We characterize the symmetry and the degeneracy of the modes. In the thermodynamic limit the Lyapunov spectrum has a diverging slope near the intersection with the abscissa. No positive lower bound exists for the positive exponents. The mode amplitude scales with the inverse square root of the particle number as expected from the normalization of the perturbation vectors. 
Christina Forster, Robin Hirschl, Harald A. Posch, William G. Hoover