Localized and Delocalized Modes in the Tangent–Space Dynamics of Planar Hard Dumbbell Fluids

Systems of hard dumbbells are, arguably, the simplest model for a molecular fluid composed of linear molecules. We study here the Lyapunov instability for two-dimensional systems containing qualitatively different degrees of freedom, translation and rotation. We characterize this instability by the Lyapunov spectrum, which measures the rate of exponential divergence, or convergence, of infinitesimal phase space perturbations along selected directions. We characterize the dependence of the spectrum and of the Kolmogorov-Sinai entropy on the density and on the dumbbell anisotropy, where the emphasis is on the thermodynamic limit. The phase space perturbation growing exponentially with a rate given by the maximum Lyapunov exponent is strongly localized in space, and this localization persists in the thermodynamic limit. The perturbations growing according to the smallest positive exponents, on the other hand, are represented by coherent wave-like structures spread out over the whole simulation box. Depending on the degeneracy of the associated exponents, these perturbations are either non-propagating transversal, or propagating longitudinal modes. Because of the analogy with the familiar hydrody-namic modes of continuum mechanics, the so-called Lyapunov modes promise to be of importance for understanding the dynamics of fluids and solids.