Calculation of the canonical ensemble

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2.3 Calculation of the canonical ensemble

Maximize

$S^{'} = - k \sum_{i} p_{i} ln (p_{i})$

(2.10)

with the restrictions

$U = \sum_{i} p_{i} U_{i} and 1 = \sum_{i} p_{i} .$

(2.11)

The restrictions are handled by Lagrange parameters $α$ and $β$ :
Variation of the function

$δ [S^{'} - k α (\sum_{i} p_{i} - 1) - k β (\sum_{i} p_{i} U_{i} - U)] = 0$

(2.12)

without restrictions leads to

$- ln (p_{i}) - 1 - α - β U_{i} = 0 .$

(2.13)

With

$\frac{1}{Z} : = exp (- 1 - α)$

(2.14)

follows

$Z (β, V, N) = \sum_{i} exp (- β U_{i}) and p_{i} = \frac{1}{Z} exp (- β U_{i}) .$

(2.15)

$Z$ is called the canonical partition function (sum of states).
We get

$U = \sum_{i} p_{i} U_{i} = \frac{\sum_{i} exp (- β U_{i}) U_{i}}{\sum_{i} exp (- β U_{i})} = - (\frac{\partial ln (Z)}{\partial β}) : = U (β, V, N)$

(2.16)

and

$S = - k \sum_{i} [\frac{1}{Z} exp (- β U_{i}) (- ln (Z) - β U_{i})] = k ln (Z) + β k U$

(2.17)

leading to:

$\begin{matrix} \begin{aligned} \frac{d S}{k} & = (\frac{\partial ln (Z)}{\partial β}) d β + (\frac{\partial ln (Z)}{\partial N}) d N + (\frac{\partial ln (Z)}{\partial V}) d V + U d β + β d U \\ = (\frac{\partial ln (Z)}{\partial N}) d N + (\frac{\partial ln (Z)}{\partial V}) d V + β d U \end{aligned} \end{matrix}$

(2.18)

This means

$S = S (V, N, U)$

(2.19)

and $S$ is the Legendre transformed of $k ln (Z)$ .
We define

$(\frac{\partial S}{\partial U}) : = \frac{1}{T} and get β = \frac{1}{k T} .$

(2.20)

Comparison of

$T S = k T ln (Z) + β k T U and F (V, N, T) = U - T S$

(2.21)

gives

$F = - k T ln (Z (V, N, T)) .$

(2.22)

In statistical mechanics the calculation of the thermodynamic potentials is transformed into the calculation of partition functions.

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