potentials, and thermo obs not yet finished

[lectures/latex.git] / posic / talks / md_simulation_von_silizium.tex

diff --git a/posic/talks/md_simulation_von_silizium.tex b/posic/talks/md_simulation_von_silizium.tex
index e0e1783..47e144a 100644 (file)
--- a/posic/talks/md_simulation_von_silizium.tex
+++ b/posic/talks/md_simulation_von_silizium.tex
@@ -39,6 +39,9 @@
 \def\slideleftmargin{5.1cm}
 \def\slidetopmargin{-0.6cm}
 
+\newcommand{\ham}{\mathcal{H}}
+\newcommand{\pot}{\mathcal{V}}
+
 % topic
 
 \begin{slide}
@@ -513,9 +516,29 @@ Problemstellung: Finden der Nachbarn f"ur Wechselwirkung
  Thermodynamische Gr"o"sen
 }
 \begin{itemize}
- \item Innere Energie: $E = ...$
- \item Temperatur
- \item Druck
+ \item Innere Energie:
+       \[
+       E = <K> + <U> = < \sum_i \frac{{\bf p}_i^2}{2m_i} > + <U({\bf q})>
+       \]
+ \item Temperatur/Druck
+       \begin{eqnarray}
+       <p_k \frac{\partial \ham}{\partial p_k}> &=& k_BT \nonumber \\
+       <q_k \frac{\partial \ham}{\partial q_k}> &=& k_BT \nonumber
+       \end{eqnarray}
+       \begin{center}
+       "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:
+       \[
+       \sum_i {\bf q}_i \nabla_{{\bf q}_i} \pot = 3Nk_BT \quad
+       \stackrel{\textrm{kart. Koord.}}{\Rightarrow} \quad 
+       - \sum_i {\bf r}_i \nabla_{{\bf r}_i} \pot = -3Nk_BT \quad
+       \]
  \item W"armekapazit"at
  \item Struktur Werte
  \item Diffusion
@@ -532,6 +555,7 @@ Problemstellung: Finden der Nachbarn f"ur Wechselwirkung
 {\large\bf
  EAM
 }
+
 \end{slide}
 
 \begin{slide}