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h1. Canonical Ensembles (fixed N, V, T)

Sum of the set of occupation numbers {<tex>a_j</tex>} is the total number of systems: 

<tex>\sum_j a_j = A</tex>

(Constraint #1)


Also, since we have an isolated system:

<tex>\sum_j a_j E_j = E </tex>

(Constraint #2)


The number of ways that any particular distribution of the {<tex>a_j</tex>} can be realized is the number of ways that <tex>A</tex> distinguishable objects can be arranged into groups (permutation):

<tex>W\raisebox{7}{\left(\left\{ a_j \right\}\right)}=\frac{A!}{a_1! a_2! a_3! .....} = \frac{A!}{\Pi_k a_k} </tex>


The fraction of systems/members of the canonical ensemble with energy <tex>E_j</tex> is <tex>\frac{a_j}{A}</tex> . The overall probability <tex>P_j</tex> that a system is in the j'th quantum state is:

<tex>P_j = \frac{\overline a_j}{A} = \frac{1}{A} \frac{\sum_a W(a)a_j(a)}{\sum_a W(a)}
</tex>

which takes the average <tex>\frac{a_j}{A}</tex> over all allowed distributions with equal a priori probabilities.


If we let <tex>A \to 0</tex> (i.e., <tex>a_j \to 0</tex>) , we get:

<tex>P_j = \frac{\overline a_j}{A} = \frac{1}{A} \frac{\sum_a W(a)a_j(a)}{\sum_a W(a)} = \frac{1}{A} \frac{W(a^*)a_j^*}{W(a^*)} = \frac{a_j^*}{A}
</tex>

where <tex>a_j^*</tex> is the value of <tex>a_j</tex> in the distribution that maximizes <tex>W(a)</tex> (i.e. it is the most probable distribution). Thus:

<tex>P_j = \frac{\overline a_j}{A} = \frac{a_j^*}{A}
</tex>

h2. Canonical Ensemble Average of any Mechanical Property

<tex>\overline M = \sum_j M_j P_j</tex>


h2. Application of Lagrange's Method of Undetermined Multipliers and Stirling's Approximation

The most probable distribution becomes:

<tex>a_j^* = e^{-\alpha' \cdot }e^{-\beta E_j}</tex>

j = 1, 2, ....

<tex>\alpha' = \alpha +1</tex>