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 {λ}, its associated eigenvectors {δ}, and the time-averaged spectrum {〈λ〉}. Each local Lyapunov exponent λ describes the degree of instability associated with a well-defined direction—given by the associated unit vector δ—in the full many-body phase space. For a variety of hard-particle systems it is by now well-established that several of the δ vectors, all with relatively-small values of the time-averaged exponent 〈λ〉, 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 〈λ〉=0.