Lyapunov modes in soft-disk systems

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.