fixed the wideset detailed balance equation

1 files changed, 6 insertions, 7 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index bf76591..02b21af 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -234,13 +234,12 @@ reverse process $P(\set{s'}\to\set s)$ by
 \[
   \begin{aligned}
     \frac{P(\set s\to\set{s'})}{P(\set{s'}\to\set s)}
-    &=\frac{f_{m_0}(r\mid\set s)}{f_{m_0}(r^{-1}\mid\set s)}\prod_{\{i,j\}\in
-  C}\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}\prod_{\{i,j\}\in\partial
-  C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}
-  =\!\!\!\!\prod_{\{i,j\}\in\partial
-  C}e^{\beta(\J(s_i',s_j')-\J(s_i,s_j))}
-  =\frac{e^{-\beta\H(\set{s'})}}{e^{-\beta\H(\set s)}},
-\end{aligned}
+    &=\frac{f(r\mid m_0,\set s)}{f(r^{-1}\mid m_0,\set s')}
+      \prod_{\{i,j\}\in }\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}
+      \prod_{\{i,j\}\in\partial C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}
+    =\!\!\!\prod_{\{i,j\}\in\partial C}e^{\beta(\J(s_i',s_j')-\J(s_i,s_j))}
+    =\frac{e^{-\beta\H(\set{s'})}}{e^{-\beta\H(\set s)}},
+  \end{aligned}
 \]
 \end{widetext}
 whence detailed balance is also satisfied.