diff --git a/Documentation/sampling.tex b/Documentation/sampling.tex index 8eafe4f1..064e09d8 100644 --- a/Documentation/sampling.tex +++ b/Documentation/sampling.tex @@ -74,7 +74,7 @@ \subsection{An explicit example of error estimation}\label{sec:autocorr} In order to determine the autocorrelation time, we calculate the correlation function \begin{equation} \label{eqn:autocorrel} - S_{\hat{O}}(t_{\textrm{Auto}})=\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\frac{\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, , + S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, , \end{equation} where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin. This function typically shows an exponential decay and the decay rate defines the autocorrelation time. %