Molecular dynamics ensemble, equation of state, and ergodicity

The variant of the NVE ensemble known as the molecular dynamics ensemble was recently redefined by Ray and Zhang [Phys. Rev. E 59, 4781 (1999)] to include the specification of a time invariant G (a function of phase and, explicitly, the time) in addition to the total linear momentum M. We reformulate this ensemble slightly as the NVEMR ensemble, in which R/N is the center-of-mass position, and consider the equation of state of the hard-sphere system in this ensemble through both the virial function and the Boltzmann entropy. We test the quasiergodic hypothesis by a comparison of old molecular dynamics and Monte Carlo results for the compressibility factor of the 12-particle, hard-disk systems. The virial approach, which had previously been found to support the hypothesis in the NVEM ensemble, remains unchanged in the NVEMR ensemble. The entropy S approach depends on whether S is defined through the phase integral over the energy sphere or the energy shell, the parameter straight theta being 0 or 1, respectively. The ergodic hypothesis is found to be supported for straight theta=0 but not for straight theta=1.