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Equilibrium ensembles

An equilibrium ensemble is one for which there is no explicit time-dependence in the phase space distribution function, tex2html_wrap_inline595 . In this case, Liouville's equation reduces to


which implies that tex2html_wrap_inline597 must be a pure function of the Hamiltonian


The specific form that tex2html_wrap_inline599 has depends on the specific details of the ensemble.

The integral over the phase space distribution function plays a special role in statistical mechanics:


It is known as the partition function and is equal to the number of members in the ensemble. That is, it is equal to the number of microstates that all give rise to a given set of macroscopic observables. Thus, it is the quantity from which all thermodynamic properties are derived.

If a measurement of a macroscopic observable tex2html_wrap_inline497 is made, then the value obtained will be the ensemble average:


Eqs. (1) and (2) are the central results of ensemble theory, since they determine all thermodynamic and other observable quantities.

Mark Tuckerman
Mon Jan 28 09:08:52 EST 2002