Numerical test of the Liouville equation

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Abstract
Dynamic ensemble theory is tested numerically for an ensemble of 1000 classical one-dimensional oscillators obeying canonical ‘‘Nosé-Hoover’’ dynamics. This dynamics couples each oscillator to a canonical heat bath characterized by a temperature and a relaxation time. Some initial oscillator conditions correspond to regular phase-space orbits of the Kolmogorov-Arnol’d-Moser torus type while others generate wider-ranging chaotic trajectories. Among the regular oscillator orbits is a set of trajectories resembling ‘‘double bedsprings,’’ with ‘‘quantized’’ values of the oscillator energy and mean-square displacement. The number which indexes these orbits corresponds to the number of coils between turning points. Despite the existence of this relatively complex mixture of regular and chaotic trajectories, the Liouville equation correctly describes phase-space flows, in both the steady equilibrium canonical-ensemble case as well as in the nonsteady cases which evolve from strongly nonequilibrium initial conditions. The source of apparent irreversibility seen in the nonsteady evolution of the oscillator ensemble is identified as a ‘‘second-law’’ attractor, usually characteristic of large thermodynamic systems. The attractor is that relatively small but highly probable portion of phase space for which observation times exceed recurrence times.

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