Canonical Ensembles (fixed N, V, T)

Sum of the set of occupation numbers

Unknown macro: {<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

can be realized is the number of ways that <tex>A</tex> distinguishable objects can be arranged into groups (permutation):

<tex>W\raisebox

Unknown macro: {7}
Unknown macro: {left(left{ a_j right}right)}

=\frac

Unknown macro: {A!}
Unknown macro: {a_1! a_2! a_3! .....}

= \frac

Unknown macro: {Pi_k a_k}

</tex>

The fraction of systems/members of the canonical ensemble with energy <tex>E_j</tex> is <tex>\frac

Unknown macro: {a_j}
Unknown macro: {A}

</tex> . The overall probability <tex>P_j</tex> that a system is in the j'th quantum state is:

<tex>P_j = \frac

Unknown macro: {overline a_j}

= \frac

Unknown macro: {1}
Unknown macro: {A}

\frac

Unknown macro: {sum_a W(a)a_j(a)}
Unknown macro: {sum_a W(a)}

</tex>

which takes the average <tex>\frac

Unknown macro: {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

Unknown macro: {overline a_j}

= \frac

Unknown macro: {1}
Unknown macro: {A}

\frac

Unknown macro: {sum_a W(a)a_j(a)}
Unknown macro: {sum_a W(a)}

= \frac

Unknown macro: {A}

\frac

Unknown macro: {W(a^*)a_j^*}
Unknown macro: {W(a^*)}

= \frac

Unknown macro: {a_j^*}

</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

Unknown macro: {overline a_j}
Unknown macro: {A}

= \frac

Unknown macro: {a_j^*}

</tex>

Canonical Ensemble Average of any Mechanical Property

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

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>

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