From: hackbard Date: Fri, 8 Jun 2007 16:27:40 +0000 (+0000) Subject: potentials X-Git-Url: https://hackdaworld.org/gitweb/?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}