Nose has developed many-body equations of motion designed to reproduce Gibbs's canonical phase-space distribution. These equations of motion have a Hamiltonian basis and are accordingly time reversible and deterministic. They include thermodynamic temperature control through a sin gle deterministic friction coefficient, which can be thought of as a control variable or as a memory function. We apply Nose's ideas to a single classical one-dimensional harmonic oscillator. This rel atively simple system exhibits both regular and chaotic dynamical trajectories, depending on the ini tial conditions. We explore here the nature of these solutions by estimating their fractal dimen sionality and Lyapunov instability. The Nose oscillator is a borderline case, not sufficiently chaotic for a fully statistical description. We suggest that the behavior of only slightly more complicated systems is considerably simpler and in accord with statistical mechanics.

Smoothed particle applied mechanics (SPAM), also referred to as smoothed particle hydrodynamics, is a Lagrangian particle method for the simulation of continuous flows. Here we apply it to the formation of a liquid drop, surrounded by its vapor, for a van der Waals (vdW) fluid in two dimensions. The cohesive pressure of the vdW equation of state gives rise to an attractive, central force between the particles with an interaction range which is assumed to exceed the interaction range of all the other smoothed forces in the SPAM equations of motion. With this assumption, stable drops are formed, and the vdW phase diagram is well reproduced by the simulations. Below the critical temperature, the surface tension for equilibrated drops may be computed from the pressure excess in their centers. It agrees very well with the surface tension independently determined from the vibrational frequency of weakly excited drops. We also study strongly deformed drops performing large-amplitude oscillations, which are reminiscent of the oscillations of a large ball of water under microgravity conditions. In an appendix we comment on the limitations of SPAM by studying the violation of angular momentum conservation, which is a consequence of noncentral forces contributed by the full Newtonian viscous stress tensor.

We show that Nose mechanics provides a link between computer simulations of nonequilibrium processes and real-world experiments. Reversible Nose equations of motion, when used to constrain nonequilibrium boundary regions, generate stable dissipative behavior within an adjoining bulk sample governed by Newton's equations of motion. Thus, irreversible behavior consistent with the second law of thermodynamics arises from completely reversible microscopic motion. Loschmidt's reversibility paradox is surmounted by this Nose-Newton system, because the steady-state nonequilibrium probability density in the many-body phase space is confined to a zero-volume attractor.

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.

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