Lyapunov modes of two-dimensional many-body systems; Soft disks, hard disks, and rotors

The dynamical instability of many-body systems can best be characterized through the local Lyapunov spectrum {lambda}, its associated eigenvectors {delta}, and the time-averaged spectrum {} P. Each local Lyapunov exponent l describes the degree of instability associated with a well-defined direction given by the associated unit vector delta-in the full many-body phase space. For a variety of hard-particle systems it is by now well-established that several of the d vectors, all with relatively-small values of the time-averaged exponent correspond to quite well-defined long-wavelength ' ' modes.' ' We investigate soft particles from the same viewpoint here, and find no convincing evidence for corresponding modes. The situation is similar no firm evidence for modes in a simple two-dimensional lattice-rotor model. We believe that these differences are related to the form of the time-averaged Lyapunov spectrum near = 0.