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\begin{equation}
H = \sum_{\substack{\langle i,j \rangle \text{ neighbours} ,\\ \sigma(i) != \sigma(j) }} J_{\sigma(i),\sigma(j)} +  \sum_{\sigma \text{ cells} } \lambda_P(P_{\sigma}-P_t)^{2}  + \sum_{\sigma \text{ cells} } \lambda_A(A_{\sigma}-A_t)^{2} 
\end{equation}

\begin{equation}
\varDelta H^\prime = \varDelta H - \sum_{\sigma \text{ cells} } \mu  (\vec{\theta_\sigma} \cdot \vec{\delta C_\sigma})
\end{equation}

\begin{equation}\label{eq:Metropolis}
 p =
  \begin{cases}
    1 & \text{if } \varDelta H^\prime <0 \\
    e^{-{\frac{\varDelta H^\prime}{kT}}} & \text{if } \varDelta H^\prime \ge 0 .\\
  \end{cases}
\end{equation}



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