made correction to detailed balance equation

1 files changed, 5 insertions, 6 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index b24fa9d..2bc29ee 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -228,13 +228,12 @@ C\subset E$ is related to the probability of the reverse process $P(\set{s'}\to\
 \[
   \begin{aligned}
     \frac{P(\set s\to\set{s'})}{P(\set{s'}\to\set s)}
-    &=\frac{f_{\set s}(r\mid m_0)}{f_{\set s'}(r^{-1}\mid m_0)}\prod_{\{i,j\}\in
+    &=\frac{f_{m_0}(r\mid\set s)}{f_{m_0}(r^{-1}\mid\set s)}\prod_{\{i,j\}\in
   C}\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}\prod_{\{i,j\}\in\partial
-  C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}\\
-  &=\prod_{\{i,j\}\in\partial
-  C}e^{\beta(\J(r\cdot s_i,s_j)-\J(s_i,s_j))}
-  =\frac{e^{-\beta\H(\set
-  s)}}{e^{-\beta\H(\set{s'})}},
+  C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}
+  =\prod_{\{i,j\}\in\partial
+  C}e^{\beta(\J(s_i',s_j')-\J(s_i,s_j))}
+  =\frac{e^{-\beta\H(\set{s'})}}{e^{-\beta\H(\set s)}},
 \end{aligned}
 \]
 \end{widetext}