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-rw-r--r--monte-carlo.tex13
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.