X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=ising%2Fising_slides.tex;h=29a79263ba9d8f97208bce6eb4c04e872f2e3894;hp=a52fed78fa4cb851375c94eec57b39e2f9ff9faa;hb=HEAD;hpb=68c63b155821d71e6b470719efae7b8d99bce6f7

diff --git a/ising/ising_slides.tex b/ising/ising_slides.tex
index a52fed7..29a7926 100644
--- a/ising/ising_slides.tex
+++ b/ising/ising_slides.tex
@@ -128,7 +128,10 @@ Molekularfeldn"aherung:\\
 Approximation: Vernachl"assigung der Spinfluktuationen $S_i-<S_i>$\\
 Spin-Wechselwirkungs-Term:
 \[
- S_iS_j = (S_i-m+m)(S_j-m+m)=m^2+m(S_i-m)+m(S_j-m)+(S_i-m)(S_j-m)
+\begin{array}{ll}
+ S_iS_j & = (S_i-m+m)(S_j-m+m)\\
+        & = m^2+m(S_i-m)+m(S_j-m)+(S_i-m)(S_j-m)
+\end{array}
 \]
 wobei:
 \begin{itemize}
@@ -169,8 +172,8 @@ implizite Bestimmungsgleichung f"ur $B_0=0$:
 
 \begin{slide}
 \begin{itemize}
-\item L"osung: $m \neq 0 \longleftrightarrow \frac{\partial (\tanh (\beta Jm))}{\partial m} > 1$ bei $m=0$
-\item kritische Temperatur: $\frac{\partial (\tanh (\beta Jm))}{\partial m} = 1$ bei $m=0$
+\item L"osung: $m \neq 0 \longleftrightarrow \frac{\partial (\tanh (\beta Jm))}{\partial m} > 1$
+\item kritische Temperatur: $\frac{\partial (\tanh (\beta Jm))}{\partial m} = 1$ f"ur $m=0$
 \end{itemize}
 % \setlength{\unitlength}{2cm}
 % \begin{picture}(6,4)(-3,-2)
@@ -282,7 +285,7 @@ Die Matrix muss also wie folgt aussehen:
 Zustandssumme:
 \[
  \begin{array}{ll}
- \displaystyle Z & \displaystyle = \sum_{S_1} \sum_{S_2} \ldots \sum_{S_N} <S_1|\mathbf{T}|S_2> <S_2|\mathbf{T}|S_3> \ldots <S_{N-1}|\mathbf{T}|S_N> <S_N|\mathbf{T}|S_1> \\[2mm]
+ \displaystyle Z & \displaystyle = \sum_{S_1} \sum_{S_2} \ldots \sum_{S_N} <S_1|\mathbf{T}|S_2> <S_2|\mathbf{T}|S_3> \ldots <S_N|\mathbf{T}|S_1> \\[2mm]
  \displaystyle  & \displaystyle = \sum_{S_1} <S_1|\mathbf{T}^N|S_1> \\[2mm]
  \displaystyle  & \displaystyle = \textrm{Sp} \, \mathbf{T}^N
  \end{array}
@@ -305,7 +308,7 @@ F"ur den Fall $B_0 = 0$ gilt:
 \[
  \begin{array}{l}
   \displaystyle \lambda_{\pm} = e^K \pm e^{-K} \\[2mm]
-  \displaystyle Z = 2^N \cosh^N (K) + 2^N \sinh^N (K) = 2^N \cosh^N (K) (1 + \tanh^N (K)) \stackrel{N >> 1}{\longrightarrow} 2^N \cosh^N (K) \\[2mm]
+  \displaystyle Z = 2^N \cosh^N (K) + 2^N \sinh^N (K) \stackrel{N >> 1}{\longrightarrow} 2^N \cosh^N (K) \\[2mm]
   \displaystyle F = -k_B T \, \textrm{ln} \, Z \stackrel{N >> 1}{\longrightarrow} -N k_B T \, \textrm{ln} \, (2 \cosh (\beta J))
  \end{array}
 \]
@@ -349,7 +352,7 @@ Erkenntnis:
 \end{itemize}
 F"ur $T=0$:
 \[ 
- \lim_{T \rightarrow 0} \frac{\lambda_+}{\lambda_-}=1 \, \textrm{obere Approximation nichtmehr g"ultig)}
+ \lim_{T \rightarrow 0} \frac{\lambda_+}{\lambda_-}=1 \, \textrm{, ist obere Approximation nichtmehr g"ultig}
 \]
 Phasen"ubergang bei $B_0=T=0$ (Korrelationsl"ange geht gegen unendlich)\\
 Kritische Exponenten:
@@ -497,7 +500,7 @@ somit gilt:
 
 \begin{slide}
 Realisierung einer Boltzmannverteilung: Metropolis Algorithmus\\
-$[4]$ http://www.npac.syr.edu/users/gcf/slitex/CPS713MonteCarlo96/index.html
+$[4]$ http://www.npac.syr.edu/users/gcf/slitex/CPS713MonteCarlo96/
 \[
  W(A \rightarrow B) = \left\{
  \begin{array}{ll}
@@ -608,7 +611,7 @@ Spingl"aser: Optimierung und Ged"achtnis
 \item \label{lit1} W. Nolting, Grundkurs: Statistische Physik, Band 6
 \item \label{lit2} Rodney J. Baxter, Exactly Solved Models in Statistical Mechanics
 \item \label{lit3} http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture\_26/node2.html
-\item \label{lit4} http://www.npac.syr.edu/users/gcf/slitex/CPS713MonteCarlo96/index.html
+\item \label{lit4} http://www.npac.syr.edu/users/gcf/slitex/CPS713MonteCarlo96/
 \item \label{lit5} Hildegard Meyer-Ortmanns, Abstract: Immigration, integration and ghetto formation
 \item \label{lit6} K. Malarz, Abstract: Social phase transition in Solomon network
 \item \label{lit7} Kerson Huang, Statistical mechanics