Commit fbf2fa89 authored by Michael Wimmer's avatar Michael Wimmer
Browse files

fix math

parent 60704d66
Pipeline #58155 passed with stages
in 1 minute and 33 seconds
......@@ -79,12 +79,12 @@ case we could instead try to sample from a nearby distribution $p_\text{sample}(
In this way we can make sure to approximately focus on the physically relevant configurations.
Doing this requires us to rewrite the integral as
$$\begin{split}
$$\begin{eqnarray}
\int p_\text{real}(R) A(R) dR = &
\int p_\text{sampling}(R) \underbrace{\frac{p_\text{real}(R)}{p_\text{sampling}(R)}}}_{=w(R)} A(R) dR\\
\int p_\text{sampling}(R) \underbrace{\frac{p_\text{real}(R)}{p_\text{sampling}(R)}}_{=w(R)} A(R) dR\\
= & \int p_\text{sampling}(R) w(R) A(R) dR\\
\approx & \frac{1}{N} \sum_{i=1}^N w(R_i) A(R_i) \tag{3}
\end{split}$$
\end{eqnarray}$$
where the configurations $R_i$ are now sampled from $p_\text{sampling}(R)$. When using this approximate probability distribution
we thus have to introduce *weights* $w(R)$ into the average.
......
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