We consider simulations of a two-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing Lyapunov exponents. To this spectrum are associated "eigen-directions", called Lyapunov modes. We carefully analyze these modes and show how they are naturally associated with vector fields over the container. We also show that the Lyapunov exponents, and the coupled dynamics of the modes (where it exists) follow linear laws, whose coefficients only depend on the density of the gas, but not on aspect ratio and very little on the boundary conditions.

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.

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Lyapunov modes are periodic spatial perturbations of phase-space states of many-particle systems, which are associated with the small positive or negative Lyapunov exponents. Although familiar for hard-particle systems in one, two, and three dimensions, they have been difficult to find for soft-particles. We present computer simulations for soft-disk systems in two dimensions and demonstrate the existence of the modes, where also Fourier-transformation methods are employed. We discuss some of their properties in comparison with equivalent hard-disk results. The whole range of densities corresponding to fluids is considered. We show that it is not possible to represent the modes by a two-dimensional vector field of the position perturbations alone (as is the case for hard disks), but that the momentum perturbations are simultaneously required for their characterization.For the last 50 years molecular dynamics simulations have decisively contributed to our understanding of the structure and dynamics of simple fluids and solids [1]. More recently, also the concepts of dynamical systems theory have been applied to study the tangentspace dynamics of such systems [2]. Of particular interest is the extreme sensitivity of the phase-space evolution to small perturbations. On average, such perturbations grow, or shrink, exponentially with time, which may be characterized by a set of rate constants, the Lyapunov exponents. The whole set of exponents is referred to as the Lyapunov spectrum.This instability is at the heart of the ergodic and mixing properties of a fluid and offers a new tool for the study of the microscopic dynamics. In particular, it was recognized very early that there is a close connection with the classical transport properties of systems in nonequilibrium stationary states [3]. For fluids in thermodynamic equilibrium, an analysis of the Lyapunov instability is expected to provide an unbiased expansion of the dynamics into events, which, in favorable cases, may be associated with qualitatively different degrees of freedom, such as the translation and rotation of linear molecules [4], or with the intramolecular rotation around specific chemical bonds [5].Since the pioneering work of Bernal [6] with steel balls, hard disks have been considered the simplest model for a "real" fluid. With respect to the structure, they serve as a reference system for highly-successful perturbation theories of liquids [7,8]. Recently we studied the Lyapunov instability of such a model and found [9,10,11,12,13] 1. that the slowly-growing and decaying perturbations associated with the non-vanishing Lyapunov exponents closest to zero may be represented as periodic vector fields coherently spread out over the physical space and with well-defined wave vectors k.Because of their similarity with the classical modes of fluctuating hydrodynamics we refer to them as Lyapunov modes. Depending on the boundary conditions, the respective exponents are degenerate, and the spectrum has a step-like appearance in that regime.2. that the fast-gr...

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