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<math>\frac{P_{\lbrace\sigma\rbrace\rightarrow\lbrace\sigma'\rbrace}}{P_{\lbrace\sigma'\rbrace\rightarrow\lbrace\sigma\rbrace}}=\frac{Pr\left(\lbrace\sigma'\rbrace|B.C.\right)Pr\left(B.C.|\lbrace\sigma\rbrace\right)}{Pr\left(\lbrace\sigma\rbrace|B.C.\right)Pr\left(B.C.|\lbrace\sigma'\rbrace\right)}=\frac{p\cdot \exp\left[-2\beta\sum\limits_{<l,m>}\delta_{\sigma_l,\sigma_m}J_{lm}\right]}{p\cdot \exp\left[-2\beta\sum\limits_{<l,m>}\delta_{\sigma'_l,\sigma'_m}J_{lm}\right]}
=
since <math>\Delta E=-\sum\limits_{<l,m>}J_{lm}\left(\sigma'_l \sigma'_m - \sigma_l \sigma_m\right)=-\sum\limits_{<l,m>}J_{lm}\left[\delta_{\sigma'_l,\sigma'_m}-\left(1-\delta_{\sigma'_l,\sigma'_m}\right)-\delta_{\sigma_l,\sigma_m}+\left(1-\delta_{\sigma_l,\sigma_m}\right)\right]=-2\sum\limits_{<l,m>}J_{lm}\left(\delta_{\sigma'_l,\sigma'_m}-\delta_{\sigma_l,\sigma_m}\right)</math>.
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