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@@ -593,27 +593,34 @@ which we can rewrite as a functional derivative of the ground state energy per-p
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C_2(\mathbf x)=2\rho\frac{\delta e_0}{\delta v(\mathbf x)}
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.
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\end{equation}
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The prediction $\tilde C_2$ of the Big, Medium and Simple Equations for the two-point correlation function are therefore defined by differentiating $\tilde e$ in\-~(\ref{erel}) with respect to $v$:
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The prediction $\tilde C_2$ of the Big and Medium Equations for the two-point correlation function are therefore defined by differentiating $\tilde e$ in\-~(\ref{erel}) with respect to $v$:
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\begin{equation}
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\tilde C_2(\mathbf x):=2\rho\frac{\delta\tilde e}{\delta v(\mathbf x)}
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.
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\label{C2}
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\end{equation}
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\indent
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In the case of the simple equation, we will proceed differently.
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If we were to define $\tilde C_2$ as in\-~(\ref{C2}), we would find that $\tilde C_2$ would not converge to $\rho^2$ as $|\mathbf x|\to\infty$, which is obviously unphysical.
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This comes from the fact that first approximating $S$ as in\-~(\ref{approx1}) and then differentiating with respect to $v$ is less accurate than first differentiating with respect to $v$ and then approximating $S$.
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Defining $\tilde C_2$ following the latter prescription, we find that, for the Simple Equation,
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\begin{equation}
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\begin{array}{>\displaystyle l}
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\tilde C_2(\mathbf x)=
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\rho^2\tilde g_2(\mathbf x)+
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\\[0.3cm]
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+\rho^2\frac{\mathfrak K_{\tilde e}v(\mathbf x)\tilde g_2(\mathbf x)-2\rho u\ast \mathfrak K_{\tilde e}v(x)+\rho^2u\ast u\ast \mathfrak K_{\tilde e}v(x)}{1-\rho\int d\mathbf x\ v(\mathbf x)\mathfrak K_{\tilde e}(2u(\mathbf x)-\rho u\ast u(\mathbf x))}
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\end{array}
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\label{correlation_simpleq}
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\end{equation}
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where $\mathfrak K_{\tilde e}$ is the operator defined in\-~(\ref{Ke}).
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Defined in this way, $\tilde C_2\to\rho^2$ as $|\mathbf x|\to\infty$.
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\bigskip
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\indent
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$C_2$ is the physical correlation function, using the probability distribution $|\psi_0|^2$, but, as we saw in section\-~\ref{sec:approx}, $\psi_0$ can also be thought of a probability distribution, whose two-point correlation function is $g_2$, defined in\-~(\ref{g}).
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The Big, Medium and Simple Equations make a natural prediction for the function $g_2$: namely $1-u(\mathbf x)$.
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In the case of the Simple Equation, we can directly relate $\tilde C_2$ and $\tilde g_2\equiv1-u(\mathbf x)$:
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\begin{equation}
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\tilde C_2(\mathbf x)=\rho^2\frac{(1-\mathfrak K_{\tilde e}v(\mathbf x))\tilde g_2(\mathbf x)}{1-\rho\int d\mathbf x\ v(\mathbf x)\mathfrak K_{\tilde e}(2u(\mathbf x)-\rho u\ast u(\mathbf x))}
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\label{correlation_simpleq}
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\end{equation}
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where $\mathfrak K_{\tilde e}$ is the operator defined in\-~(\ref{Ke}).
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We have shown in\-~\cite{CJL20b} that $\mathfrak K_{\tilde e} v$ behaves like $|\mathbf x|^{-2}$ as $|\mathbf x|\to\infty$, whereas $u\equiv1-\tilde g_2$ goes like $|\mathbf x|^{-4}$.
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In particular, this means that the $|x|\to\infty$ limit of $\tilde C_2$ is $\rho^2/(1-\rho\int d\mathbf x\ v(\mathbf x)\mathfrak K_{\tilde e}(2u(\mathbf x)-\rho u\ast u(\mathbf x)))$, whereas it is simply $\rho^2$ for the exact ground state of the Bose gas.
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This means that the prediction of the simple equation is only accurate in the $\rho\to0$ limit, in which the denominator in\-~(\ref{correlation_simpleq}) tends to 1.
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In addition, the truncated correlation function decays like $|\mathbf x|^{-2}$, whereas the prediction for the Bose gas\-~\cite{LHY57} is that it should decay as $|\mathbf x|^{-4}$.
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However, we can show that $\mathfrak K_{\tilde e}v$ is of a higher order in $\rho$ compared to $u$, so in the $\rho\to0$ limit, the truncated correlation function decays like $u$, and the simple equation recovers the $|\mathbf x|^{-4}$ decay predicted for the Bose gas.
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The Big, Medium and Simple Equations make a natural prediction for the function $g_2$: namely $1-u(\mathbf x)$.
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\bigskip
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\indent
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@@ -640,7 +647,7 @@ Note that this behavior was observed for the stronger potential $16e^{-|\mathbf
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\indent
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In Figure\-~\ref{fig:correlation}, we compare the prediction $\tilde C_2$ to the QMC simulation.
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At low densities, the prediction of the Big Equation agrees rather well with the QMC simulation.
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The Simple and Medium Equations are not as accurate; in particular, for the Simple Equation, $\tilde C_2$ does not tend to $\rho^2$ as $|\mathbf x|\to\infty$, as can be seen from\-~(\ref{correlation_simpleq}).
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The Simple and Medium Equations are not as accurate.
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At larger densities, the Simple and Medium Equations are quite far from the QMC computation, and the Big Equation is not as accurate as in the case of $\tilde g_2$, but it does reproduce some of the qualitative behavior of the QMC computation.
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In particular, there is a local maximum in the two-point correlation function, which occurs at a length scale that is close to that observed for $\tilde g_2$.
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At small $\mathbf x$, $\tilde C_2$ is negative, which is clearly not physical, and those values should be discarded.
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