Relationship of Dynamical and Canonical-Ensemble Pressures

In order to relate the dynamical pressure (29) to the canonical-ensemble pressure (8), we compute 

By the quasi-ergodic hypothesis we replace (W nvem by W and determine the dependence of W on H and M as follows. From the conservation of momentum, one has 

Thus W t)y Eq. (29), depends on relative coordinates and velocities only and therefore for fixed N, V, r, and is independent of M. Moreover, the hard-core trajectory scales with the magnitude of the velocity as 

Evaluating (E) = (1 — l/N)(E) and replacing fi(n, N) by the dynamical pressure, we find the canonical-ensemble pressure from (8) to be 

Hoover and Alder found the Monte Carlo and molecular dynamics results for the equation of state of systems of 12 hard disks to be consistent with (40) to within statistical error. 

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