From 543cfe0dc2284bb3b27e26d959d64021519a8e0e Mon Sep 17 00:00:00 2001
From: hackbard <hackbard>
Date: Sat, 2 Jul 2005 02:17:47 +0000
Subject: [PATCH] the talk as i gave it on 30.06.2005 - will be improved!!!

---
 computational_physics/cp.tex | 41 +++++++++++++++++++-----------------
 1 file changed, 22 insertions(+), 19 deletions(-)

diff --git a/computational_physics/cp.tex b/computational_physics/cp.tex
index f961215..f112233 100644
--- a/computational_physics/cp.tex
+++ b/computational_physics/cp.tex
@@ -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}}
-- 
2.39.2