From Eq. (2.45) we know the grand canonical potential
|
| \begin{equation*} \Omega(T,V,\mu) = \pm k T \sum_{\alpha} \ln\left( 1 \mp \exp\left(- \frac{\epsilon_{\alpha}- \mu}{k T} \right) \right) \label{gc_pot} \end{equation*} | (3.4) |
\(\alpha\) indicates independent states, the plus/minus sign depend on the particles to be Fermion or Bosons.
For a continuous system the sum changes into an integral:
Let
|
| \begin{equation*} F_{\beta\mu}(\epsilon):= \pm k T \ln\left( 1 \mp \exp\left(- \frac{\epsilon- \mu}{k T} \right) \right) \end{equation*} | (3.5) |
and
Obviously we find
|
| \begin{equation*} \Omega(T, V, \mu) = \int_{-\infty}^{+\infty} D(\epsilon)F_{\beta\mu}(\epsilon) d\epsilon \qquad , \label{gc_pot_2} \end{equation*} | (3.7) |
|
| \begin{equation*} N(T, V, \mu) = \int_{-\infty}^{+\infty} D(\epsilon)f_{\beta\mu}(\epsilon) d\epsilon \qquad , \end{equation*} | (3.8) |
and
|
| \begin{equation*} E(T, V, \mu) = \int_{-\infty}^{+\infty} \epsilon D(\epsilon)f_{\beta\mu}(\epsilon) d\epsilon \qquad . \label{gc_energy} \end{equation*} | (3.9) |
\(D(\epsilon)\) is the density of states.
|
|
|
|
© J. Carstensen (Stat. Meth.)