author hackbard Sat, 2 Jul 2005 02:17:47 +0000 (02:17 +0000) committer hackbard Sat, 2 Jul 2005 02:17:47 +0000 (02:17 +0000)

index f961215..f112233 100644 (file)
@@ -186,7 +186,7 @@ $\Rightarrow$ study and implementation of numerical algorithms
$x^1 = x_0;$ & $y^1 = y_0;$ & \\
$v^1_x = v_{x_0};$ & $v^1_y = v_{y_0};$ & \\
loop: & $x^2 = x^1 + \tau v^1_x;$ & $y^2 = y^1 + \tau v^1_y;$ \\
-               & $v^2_x = v^1_x;$ & $v^2_y = v^1_y + (-mg) \tau;$ \\
+               & $v^2_x = v^1_x;$ & $v^2_y = v^1_y - g \tau;$ \\
& $x^1 = x^2;$ & $y^1 = y^2$ \\
& $v^1_x = v^2_x;$ & $v^1_y = v^2_y;$ \\
\end{tabular}
@@ -271,7 +271,7 @@ division by modulus $\Rightarrow$ uniform deviates : \\
\begin{itemstep}
\item transformation method:
\begin{itemize}
-            \item arbitrary propability distribution $\rho(y)$
+            \item arbitrary probability distribution $\rho(y)$
\item trafo: $p(x) dx = \rho(y) dy \Rightarrow x = \int_{- \infty}^y \rho(y) dy$
\item get inverse of $x(y) \Rightarrow y(x)$
\end{itemize}
@@ -337,35 +337,38 @@ Z = \sum_{i=1}^N e^{\frac{-E_i}{k_B T}} = Tr(e^{-\beta H})
}
\end{slide}}

-\overlays{4}{
+\overlays{2}{
\begin{slide}{metropolis algorithm}
\begin{itemstep}
\item importance sampling: \\
$<A> = \sum_i p_i A_i \approx \frac{1}{N} \sum_{i=1}^N A_i$ , with \$6pt] - \qquad p_i = \frac{e^{\beta E_i}}{Z} - \item markov process: \\ - \begin{itemize} - \item P(A,t): probability of configuration A at time t - \item W(A \rightarrow B): transition probability - \[ - \begin{array}{l} - P(A,t+1) = P(A,t) + \\ - \sum_B \Big( W(B \rightarrow A) P(B,t) - W(A \rightarrow B) P(A,t) \Big) - \end{array} -$
-          \end{itemize}
+          $\qquad p_i = \frac{e^{- \beta E_i}}{Z}$
+    \item detailed balance \$6pt] + sufficient condition for equilibrium: \\ + \[ + W(A \rightarrow B) p(A) = W(B \rightarrow A) p(B) +$
+          $\Rightarrow \frac{W(A \rightarrow B)}{W(B \rightarrow A)} = \frac{p(B)}{p(A)} = e^{\frac{- \Delta E}{k_B T}}$ \$6pt] + with \Delta E = E(B) - E(A) \end{itemstep} \end{slide}} \overlays{5}{ \begin{slide}{metropolis algorithm} \begin{itemstep} - \item detailed balance + \item choose W: \\ + \[ + W(A \rightarrow B) = \left\{ + \begin{array}{ll} + e^{- \beta \Delta E} & : \Delta E > 0 \\ + 1 & : \Delta E < 0 + \end{array} \right. +$
\item algorithm:
\begin{itemize}
-            \item visit every lattice site
-            \item calculate $\delta E$ for spin flip
-            \item flip spin if $r \leq e^{\frac{-\delta E}{k_B T}}$
+             \item visit every lattice site
+             \item calculate $\Delta E$ for spin flip
+             \item flip spin if $r \leq e^{\frac{-\Delta E}{k_B T}}$
\end{itemize}
\end{itemstep}
\end{slide}}