Here we could repeat the calculation of the last section just with one less restriction, since for all micro states \(N_i = N\) holds. This simplification we can take directly into account in the grand canonical partition function of Eq. (5.46)
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| \begin{equation*} Z_{GC} = \sum_i \exp(-\beta U_i - \gamma N_i) = \sum_i \exp(-\beta U_i - \gamma N)= \exp(- \gamma N) \sum_i \exp(-\beta U_i):= \exp(- \gamma N) Z_{C} \quad , \end{equation*} | (5.56) |
with the definition of the canonical partition function
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| \begin{equation*} Z_{C} = \sum_i \exp(-\beta U_i) \quad . \end{equation*} | (5.57) |
This result we can directly incorporate into Eq. (5.55)
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| \begin{equation*} \Omega(T,V,\mu) = -k T \ln(Z_C(T, V, \mu)) - k T \ln(\exp(- \gamma N)) = -k T \ln(Z_C(T, V, \mu)) - \mu N \qquad . \label{eq_Omega_GC_3} \end{equation*} | (5.58) |
Comparison with Eq. (5.54) directly gives
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| \begin{equation*} -k T \ln(Z_C) = U - T S = F \qquad . \label{eq_F_C_1} \end{equation*} | (5.59) |
Of course the variable \(\mu\) has to be replaced by its dependence of \(N\) leading to the standard definition of the free energy \(F(T,V,N)\) and showing that \(\Omega\) and \(F\) are Legendre transformed with respect to the pair \((N \leftrightarrow \mu)\).
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© J. Carstensen (TD Kin II)