+ \item Innere Energie:
+ \[
+ E = <K> + <U> = < \sum_i \frac{{\bf p}_i^2}{2m_i} > + <U({\bf q})>
+ \]
+ \item Temperatur/Druck
+ \[
+ <p_k \frac{\partial \ham}{\partial p_k}> = k_BT, \quad
+ <q_k \frac{\partial \ham}{\partial q_k}> = k_BT
+ \]
+ \begin{center}
+ {\em "Aquipartitionstheorem}
+ \end{center}
+ Temperatur:
+ \[
+ <\sum_i {\bf p}_i \frac{{\bf p}_i}{m_i}> = 3Nk_BT \quad
+ \Rightarrow \quad T=\frac{1}{3Nk_B} <\sum_i \frac{{\bf p}_i^2}{m_i}>
+ \]
+ Druck:
+ \[
+ <\sum_i {\bf q}_i \nabla_{{\bf q}_i} \foo> = 3Nk_BT \quad
+ \stackrel{\textrm{kart. Koord.}}{\Rightarrow} \quad
+ - \frac{1}{3} <\sum_i {\bf r}_i \nabla_{{\bf r}_i} \foo> = -Nk_BT
+ \]
+ \begin{center}
+ mit
+ \end{center}
+ \[
+ - \nabla_{{\bf r}_i} \foo = {\bf f}_i^{tot} = {\bf f}_i^{ext} + {\bf f}_i^{int}
+ \]
+ \begin{center}
+ wobei
+ \end{center}
+ \[
+ \frac{1}{3} \sum_i {\bf r}_i {\bf f}_i^{ext}=-pV, \quad
+ \frac{1}{3} \sum_i {\bf r}_i {\bf f}_i^{int}=
+ - \frac{1}{3} \sum_i {\bf r}_i \nabla_{{\bf r}_i} \pot = \vir
+ \]
+ \begin{center}
+ folgt
+ \end{center}
+ \[
+ pV = Nk_BT + <\vir>
+ \]
+\end{itemize}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Thermodynamische Gr"o"sen
+}
+\begin{itemize}