Recently, a new algorithm for the computation of covariant Lyapunov vectors and of corresponding local Lyapunov exponents has become available. Here we study the properties of these still unfamiliar quantities for a simple model representing a harmonic oscillator coupled to a thermal gradient with a two-stage thermostat, which leaves the system ergodic and fully time reversible. We explicitly demonstrate how time-reversal invariance affects the perturbation vectors in tangent space and the associated local Lyapunov exponents. We also find that the local covariant exponents vary discontinuously along directions transverse to the phase flow.

We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard-disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes parallel to the x-axis of the box. The Oseledec splitting into covariant subspaces of the tangent space is considered by computing the full set of covariant perturbation vectors co-moving with the flow in tangent space. These vectors are shown to be transversal, but generally not orthogonal to each other. Only the angle between covariant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the probability of this angle to vanish approaches zero. The stable and unstable manifolds are transverse to each other and the system is hyperbolic.

The Oseledec splitting of the tangent space into covariant subspaces for a hyperbolic dynamical system is numerically accessible by computing the full set of covariant Lyapunov vectors. In this paper, the covariant Lyapunov vectors, the orthogonal Gram-Schmidt vectors, and the corresponding local (time-dependent) Lyapunov exponents, are analyzed for a planar system of rough hard disks (RHDS). These results are compared to respective results for a smooth-hard-disk system (SHDS). We find that the rotation of the disks deeply affects the Oseledec splitting and the structure of the tangent space. For both the smooth and rough hard disks, the stable, unstable and central manifolds are transverse to each other, although the minimal angle between the unstable and stable manifolds of the RHDS typically is very small. Both systems are hyperbolic. However, the central manifold is precisely orthogonal to the rest of the tangent space only for the smooth-particle case and not for the rough disks. We also demonstrate that the rotations destroy the Hamiltonian character for the rough-hard-disk system. ‡ Present address:

The stability of binary fluid mixtures, with respect to a demixing transition, is examined within the framework of the geometrical approximation of the direct correlation for hard nonspherical particles. In this theory, the direct correlation function is essentially written in terms of the geometrical properties of the individual molecules, and those of the overlap region between two different molecules, taken at fixed separation and orientations. Within the present theory, the demixing spinodal line in the (rho(1),rho(2)) concentration plane is obtained analytically, and shown to be a quadratic function of the total packing fraction and the compositions. The theory is applied herein to binary mixtures of hard spherocylinders in the isotropic phase. Isotropic fluid-fluid demixing can be predicted for a large variety of sizes and aspect ratios, and the necessary condition for entropic demixing is a sufficiently large thickness difference between the two particles that belong to each of the fluids in the mixture. As the theory reduces exactly to the Percus-Yevick approximation for a hard sphere mixture, accordingly it will not predict fluid-fluid demixing for this particular case. Demixing is also forbidden in two other cases; for a mixture of spherocylinders and small spheres, and for mixtures of equally thin spherocylinders. The influence and competition of an ordering instability on the demixing is also examined. The ordering of a fluid will always be displaced toward higher packing fractions by the addition of a nonordering fluid, and in some cases the entropic demixing can dominate the entire fluid range. Although the present theory merges exactly with the correct Onsager limit, it is shown that, for intermediate cases, the results can be significantly different from predictions of Onsager type approaches. These discrepancies are analyzed in particular for the needle plus spherocylinder mixture. Finally, in view of the nature of the theory, it is conjectured that the predicted demixing densities values are rather upper bounds to what should be expected.