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\langle A \rangle   =  \frac{ \int A(\vec{r})
     \exp(-\frac{E(\vec{r})}{k_BT}) d\vec{r})}
     {\int \exp(-\frac{E(\vec{r})}{k_BT}) d\vec{r}}   
       =   \frac{\sum_{m} A_m}{\sum_{m} 1} 
      =  \frac{1}{M} \sum_m A_m
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