From: hackbard <hackbard>
Date: Fri, 8 Jun 2007 16:27:40 +0000 (+0000)
Subject: potentials
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=d8ba9d1112e8f769a1c0043262e1239d0ee8a984;p=lectures%2Flatex.git

potentials
---

diff --git a/posic/talks/md_simulation_von_silizium.tex b/posic/talks/md_simulation_von_silizium.tex
index b584142..d2b37f4 100644
--- a/posic/talks/md_simulation_von_silizium.tex
+++ b/posic/talks/md_simulation_von_silizium.tex
@@ -197,25 +197,173 @@ $\Rightarrow$ observable Gr"o"sen durch zeitlichen Mittelwert\\
 
 \begin{slide}
 {\large\bf
+ {\em Predictor-Corrector} Algorithmus
+}
+\begin{itemize}
+ \item Vorhersage der Orte, Geschwindigkeiten, Beschleunigungen etc ...
+       \begin{eqnarray}
+       {\bf r}^p(t + \delta t) &=& {\bf r}(t) + \delta t {\bf v}(t) +
+                                \frac{1}{2} \delta t^2 {\bf a}(t) +
+                                \frac{1}{6} \delta t^3 {\bf b}(t) + \ldots
+                                \nonumber \\
+       {\bf v}^p(t + \delta t) &=& {\bf v}(t) + \delta t {\bf a}(t) +
+                                \frac{1}{2} \delta t^2 {\bf b}(t) + \ldots
+                                \nonumber \\
+       {\bf a}^p(t + \delta t) &=&{\bf a}(t) + \delta t {\bf b}(t) + \ldots
+                                \nonumber \\
+       {\bf b}^p(t + \delta t) &=&{\bf b}(t) + \ldots
+                                \nonumber
+       \end{eqnarray}
+ \item Brechnung der tats"achlichen Kraft/Beschleunigung ${\bf a}^c$
+       f"ur die vorhergesagten Orte ${\bf r}^p$ \\
+       $\Rightarrow$ Korrekturfaktor:
+       $\Delta {\bf a}(t + \delta t) =
+       {\bf a}^c(t + \delta t) - {\bf a}^p(t + \delta t)$
+ \item Korrektur:
+       \begin{eqnarray}
+       {\bf r}^c(t + \delta t) &=& {\bf r}^p(t + \delta t) +
+                                   c_0 \Delta {\bf a}(t + \delta t) \nonumber \\
+       {\bf v}^c(t + \delta t) &=& {\bf v}^p(t + \delta t) +
+                                   c_1 \Delta {\bf a}(t + \delta t) \nonumber \\
+       {\bf a}^c(t + \delta t) &=& {\bf a}^p(t + \delta t) +
+                                   c_2 \Delta {\bf a}(t + \delta t) \nonumber \\
+       {\bf b}^c(t + \delta t) &=& {\bf b}^p(t + \delta t) +
+                                   c_3 \Delta {\bf a}(t + \delta t) \nonumber
+       \end{eqnarray}
+ \item Optional: Iteration des Korrekturschrittes
+\end{itemize}
+{\scriptsize
+ C. W. Gear.
+ The numerical integration of ordinary differential equations of various orders.
+ (1966)\\
+ C. W. Gear.
+ Numerical initial value problems in ordinary differential equations.
+ (1971)
+}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Velocity Verlet
+}\\
+Aus formaler L"osung der Liouville-Gleichung f"ur Ensemble Zeitentwicklung:
+\begin{eqnarray}
+ {\bf r}(t+\delta t) &=& {\bf r}(t) + \delta t {\bf v}(t) +
+                         \frac{1}{2} \delta t^2 {\bf a}(t) \nonumber \\
+ {\bf v}(t+\delta t) &=& {\bf v}(t) + \frac{1}{2} \delta t (
+                         {\bf a}(t) + {\bf a}(t+\delta t)) \nonumber
+\end{eqnarray}
+Alogrithmus:
+\begin{itemize}
+ \item Berechnung der neuen Ortskoordinaten ${\bf r}(t+\delta t)$
+ \item Erste Berechnung der Geschwindigkeiten
+       \[
+       {\bf v}(t+\delta t/2) = {\bf v}(t) + \frac{1}{2} \delta t {\bf a}(t)
+       \]
+ \item Berechnung der Kr"afte f"ur die Orte ${\bf r}(t+\delta t)$
+       $\Rightarrow {\bf a}(t+\delta t)$
+ \item Update der Geschwindigkeiten
+       \[
+       {\bf v}(t+\delta t) = {\bf v}(t+\delta t/2) +
+                             \frac{1}{2} \delta t {\bf a}(t+\delta t)
+       \]
+\end{itemize}
+Eigenschaften:
+\begin{itemize}
+ \item entspricht {\em GEAR-3} mit Ortskorrekturfaktor $c_0=0$
+ \item einfach, schnell, wenig Speicheraufwand $(9N)$
+ \item verh"altnism"a"sig pr"azise
+\end{itemize}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Modell zur Wechselwirkung - Das Potential
+}\\
+Klassisches Potential:
+\[
+{\mathcal V} = \sum_i {\mathcal V}_1({\bf r}_i) +
+               \sum_{i,j} {\mathcal V}_2({\bf r}_i,{\bf r}_j) +
+               \sum_{i,j,k} {\mathcal V}_3({\bf r}_i,{\bf r}_j,{\bf r}_k) +
+               \ldots
+\]
+\begin{itemize}
+ \item ${\mathcal V}_1$: Eink"orperpotential (Gravitation, elektrisches Feld)
+ \item ${\mathcal V}_2$: Paarpotential
+                         (nur abh"angig vom Abstand ${\bf r}_{ij}$)
+ \item ${\mathcal V}_3$: Dreik"orperpotential
+\end{itemize}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Wahl/Kontrolle des Ensembles
+}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ kanonisches Ensemble (NVT)
+}
+\end{slide}
 
+\begin{slide}
+{\large\bf
+ isothermales isobares Ensemble (NpT)
 }
 \end{slide}
 
 \begin{slide}
 {\large\bf
+ Die Simulationszelle \& Randbedingungen
+}
+\end{slide}
 
+\begin{slide}
+{\large\bf
+ Trick: Nachbarlisten \& Zell-Methode
 }
 \end{slide}
 
 \begin{slide}
 {\large\bf
+ Thermodynamische Gr"o"sen
+}
+\end{slide}
 
+\begin{slide}
+{\large\bf
+ 3-K"orper Potentiale
 }
 \end{slide}
 
 \begin{slide}
 {\large\bf
+ Brenner / Tersoff
+}
+\end{slide}
 
+\begin{slide}
+{\large\bf
+ EAM
+}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Albe Reparametrisierung
+}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Zusammenfassung
+}
+\end{slide}
+
+\begin{slide}
+{\large\bf
+ Ausblick
 }
 \end{slide}