Hamiltonian systems with many degrees of freedom: Asymmetric motion and intensity of motion in phase space

Abstract

This paper studies the dynamics of Hamiltonian systems with many degrees of freedom by emphasizing the many-basin structure. Three-dimensional systems in general have a many-basin structure corresponding to their many stable configurations. This is in contrast to the single-basin structure of one- or two-dimensional systems including the Fermi-Pasta-Ulam model, the Lennard-Jones model, and so on. The motion of the phase point within given basins is examined depending on the type of configuration, ordered or random. The stochastic transition in which the Kol'mogorov-Arnol'd-Moser torus collapses occurs at a certain kinetic energy for ordered configurations, similar to the behavior of one- or two-dimensional systems, but not for random configurations. This suggests that the chaotic sea prevails over the phase space with random configurations. The motion of phase point among basins is described by a Bernoulli-like shift map and classified into three types of motion, depending on the magnitude of kinetic energy: types 1, 2, and 3. Furthermore, such motion is divided into two classes of motion, depending on whether or not asymmetric motion takes place. By asymmetry we mean that the motion is unidirectional to lower basins in energy. The asymmetric motion is the dynamical manifestation of ordered phases emerging from many other random ones and occurs for types 2 and 3 but not for type 1. The dynamical origin of asymmetric motion is investigated by introducing the notion of intensity of motion in phase space, which is measured by \ensuremath{\sigma}=${\mathrm{\ensuremath{\tau}}}_{\mathit{p}}$/${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ (${\mathrm{\ensuremath{\tau}}}_{\mathit{p}}$: the time scale of mixing dynamics in the momentum space; ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$: the time scale of diffusion dynamics in the coordinate space). \ensuremath{\sigma} is expressed in terms of dynamical quantities. Our assertion that two classes of motion are separated by the decisive point \ensuremath{\sigma}=1 is confirmed by performing computer simulations. The asymmetric motion is attributed to the transient formation of the canonical probabilistic measure before completion of relaxation. The notion of intensity of motion in phase space is expected to help in discovering a generation principle of ordered phases of condensed matter from many random ones. The dynamical properties of type 3 are also examined and discussed in detail. \textcopyright{} 1996 The American Physical Society.