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| -rw-r--r-- | monte-carlo.tex | 13 |
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. |