Classical Mechanical Systems
As with any ensemble of classical systems, we would like to find a corresponding probability measure on the phase space "M". This constant energy assumption means that every system in the ensemble is confined to a submanifold of phase space of constant energy "E". Call this submanifold . From the physical considerations given above, it is already clear what the probability measure on the constant energy surface ("not the full phase space") should be: namely, the trivial one that is constant everywhere. However, while only the submanifold is of interest for the microcanonical ensemble, in other, more general ensembles, it is necessary to consider the full phase space. We now construct a measure on the full phase space that is suitable for the microcanonical ensemble.
The Liouville measure on the full phase space induces a measure on in the following manner:
The measure of an open subset R of is given by
Where Q is any open subset of M such that, Q(E, E + ΔE) is part of Q with E < H < E + ΔE, and "" is the usual Liouville volume. Thus any sufficiently good (measurable) subset of can be characterized by its hyperarea(measure) with respect to .
The density function on the full phase space is the generalized function, where H is the Hamiltonian and is the hyperarea of . If Δ is a region of the phase space, the probability of a system being in a state within Δ is simply
where is the intersection of and .
Notice how one can either consider the whole phase space and use the measure whose density is a generalized function, or restrict to the constant energy surface in question and use the measure whose density is a constant function. For instance, consider a 1-dimensional harmonic oscillator. The phase space is (the position-momentum plane) and the constant energy hypersurface is the ellipse
The latter can be parametrized as
where varies between 0 and . The measure would then equal up to a constant. On the other hand, if one considers the ellipse embedded in the plane, then it would have measure zero, which is why a generalized function is used as the density.
Read more about this topic: Microcanonical Ensemble
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