Calculation of the grand canonical ensemble

5.5 Calculation of the grand canonical ensemble

Maximize

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

(5.40)

with the restrictions

\begin{equation*} 0 = \sum_i p_i - 1 \;\mbox{, and} \quad 0 = \sum_i p_i U_i - U\;\mbox{, and} \quad 0 = \sum_i p_i N_i - N \quad . \end{equation*}

(5.41)

Introducing the Lagrange parameters \(\alpha\), \(\beta\), and \(\gamma\) the 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) - k \gamma \left( \sum_i p_i N_i - N \right) \right] = 0 \end{equation*}

(5.42)

without restrictions leads to

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

(5.43)

Defining

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

(5.44)

we find

\begin{equation*} p_i = \frac{1}{Z} \exp(-\beta U_i - \gamma N_i) \qquad . \end{equation*}

(5.45)

Taking into account the first restriction one gets

\begin{equation*} Z(\beta, V, \gamma) = \sum_i \exp(-\beta U_i - \gamma N_i) \qquad . \label{eq_Z_GC} \end{equation*}

(5.46)

Finally we get

\begin{equation*} S = k \ln(Z) + \beta k U + \gamma k N \qquad . \label{S__1} \end{equation*}

(5.47)

Just by comparison we see

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

(5.48)

and

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

(5.49)

Thus the total derivative is:

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

(5.50)

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

(5.51)

and \(S\) is the Legendre transformed of \(k \ln(Z)\).
From classical thermodynamics it is well known that

\begin{equation*} dU = T dS -p dV + \mu dN \quad \mbox{, i.e.} \quad \left( \frac{\partial S}{\partial U}\right) = \frac{1}{T} \quad \mbox{, and} \quad \left( \frac{\partial S}{\partial N}\right) = - \frac{\mu}{T} \qquad . \end{equation*}

(5.52)

So by comparison with Eq. (5.50) we see

\begin{equation*} \beta = \frac{1}{kT} \quad \mbox{, and} \quad \gamma = -\frac{\mu}{kT} \qquad . \end{equation*}

(5.53)

Multiplying Eq. (5.47) with \(T\) we find

\begin{equation*} \Omega = U - \mu N- T S \label{eq_Omega_GC_1} \end{equation*}

(5.54)

and

\begin{equation*} \Omega(T,V,\mu) = -k T \ln(Z(T, V, \mu)) \qquad . \label{eq_Omega_GC_2} \end{equation*}

(5.55)