Equivalency of Ensembles

A very important aspect of the description of systems in terms of ensemble averages is that the properties of the systems should be the same as described by one or another type of ensemble. The equivalence between the description provided by the microcanonical and canonical ensembles can be demonstrated most elegantly as follows. Consider the partition function of the canonical ensemble

$$Z= \sum_k e^{-\beta E_k} = e^{-\beta A},$$ (47)

and partition the sum over states k into groups of states such that they sum up to the same energy $E_l$. Rewriting Eq. (47) according to states of equal energy $E_l$ we obtain

$$Z=\sum_l \Omega (E_l) e^{-\beta E_l}.$$ (48)

where $\Omega (E_l)$ is the number of states with energy $E_l$. Taking the continuous limit of Eq. (48) we obtain,

$$Z = \int_0^{\infty} dE \hspace{.1cm} \Omega(E) \hspace{.1cm} e^{-\beta E},$$ (49)

i.e., the partition function $Z(\beta)$ is the Laplace transform of the total number of states $\Omega(E)$ of energy $E$. Since Laplace transforms are unique, there is a one to one correspondence between $Z(\beta)$ and $\Omega(E)$ --i.e., both functions have the same information.