From: hackbard Date: Sat, 2 Jul 2005 02:17:47 +0000 (+0000) Subject: the talk as i gave it on 30.06.2005 - will be improved!!! X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=543cfe0dc2284bb3b27e26d959d64021519a8e0e;p=lectures%2Flatex.git the talk as i gave it on 30.06.2005 - will be improved!!! --- 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: \\ $ = \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}}