Calculation of the canonical ensemble

2.3 Calculation of the canonical ensemble

Maximize

\begin{equation*} S' = -k \sum_i p_i \ln(p_i) \end{equation*}

(2.10)

with the restrictions

\begin{equation*} U = \sum_i p_i U_i \quad \mbox{and} \quad 1 = \sum_i p_i \quad . \end{equation*}

(2.11)

The restrictions are handled by Lagrange parameters \(\alpha\) and \(\beta\):
Variation of the function

\begin{equation*} \delta \left[ S' - k \alpha \left( \sum_i p_i -1 \right) - k \beta \left( \sum_i p_i U_i - U \right) \right] = 0 \end{equation*}

(2.12)

without restrictions leads to

\begin{equation*} - \ln(p_i) - 1 - \alpha - \beta U_i = 0 \qquad . \end{equation*}

(2.13)

With

\begin{equation*} \frac{1}{Z} := \exp(-1-\alpha) \end{equation*}

(2.14)

follows

\begin{equation*} Z(\beta, V, N) = \sum_i \exp(-\beta U_i) \quad \mbox{and} \quad p_i = \frac{1}{Z} \exp(-\beta U_i)\qquad . \end{equation*}

(2.15)

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

\begin{equation*} U = \sum_i p_i U_i = \frac{\sum_i \exp(-\beta U_i) U_i}{\sum_i \exp(-\beta U_i)} = - \left( \frac{\partial \ln(Z)}{\partial \beta} \right):= U(\beta, V , N) \end{equation*}

(2.16)

and

\begin{equation*} S = - k \sum_i \left [ \frac{1}{Z} \exp(-\beta U_i) \left( -\ln(Z) - \beta U_i\right)\right] = k \ln(Z) + \beta k U \end{equation*}

(2.17)

leading to:

\begin{equation*} \begin{split} \frac{dS}{k} & = \left( \frac{\partial \ln(Z)}{\partial \beta} \right) d\beta + \left( \frac{\partial \ln(Z)}{\partial N} \right) dN + \left( \frac{\partial \ln(Z)}{\partial V} \right) dV + U d\beta + \beta dU\\ & = \left( \frac{\partial \ln(Z)}{\partial N} \right) dN + \left( \frac{\partial \ln(Z)}{\partial V} \right) dV + \beta dU \end{split} \end{equation*}

(2.18)

This means

\begin{equation*} S=S(V,N,U) \end{equation*}

(2.19)

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

\begin{equation*} \left( \frac{\partial S}{\partial U}\right) := \frac{1}{T} \quad \mbox{and get} \quad \beta = \frac{1}{kT} \quad . \end{equation*}

(2.20)

Comparison of

\begin{equation*} TS = kT \ln(Z) + \beta k T U \quad \mbox{and} \quad F(V,N,T) = U - TS \end{equation*}

(2.21)

gives

\begin{equation*} F = -k T \ln(Z(V,N,T)) \quad . \end{equation*}

(2.22)

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