From 201099f9d091e4cb1cf598a5470716c49b2acd3d Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Tue, 15 Dec 2020 22:22:16 +0100
Subject: v0.1

  Changed: Improved definition of the two-point correlation function for the
  Simple Equation.
---
 Carlen_Holzmann_Jauslin_Lieb_2020.tex | 31 +++++++++++++++++++------------
 1 file changed, 19 insertions(+), 12 deletions(-)

(limited to 'Carlen_Holzmann_Jauslin_Lieb_2020.tex')

diff --git a/Carlen_Holzmann_Jauslin_Lieb_2020.tex b/Carlen_Holzmann_Jauslin_Lieb_2020.tex
index 2ff6b35..3e2a478 100644
--- a/Carlen_Holzmann_Jauslin_Lieb_2020.tex
+++ b/Carlen_Holzmann_Jauslin_Lieb_2020.tex
@@ -593,27 +593,34 @@ which we can rewrite as a functional derivative of the ground state energy per-p
   C_2(\mathbf x)=2\rho\frac{\delta e_0}{\delta v(\mathbf x)}
   .
 \end{equation}
-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 $v$:
+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 $v$:
 \begin{equation}
   \tilde C_2(\mathbf x):=2\rho\frac{\delta\tilde e}{\delta v(\mathbf x)}
   .
+  \label{C2}
 \end{equation}
-\bigskip
 
 \indent
-$C_2$ 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}).
-The Big, Medium and Simple  Equations make a natural prediction for the function $g_2$: namely $1-u(\mathbf x)$.
-In the case of the Simple Equation, we can directly relate $\tilde C_2$ and $\tilde g_2\equiv1-u(\mathbf x)$:
+In the case of the simple equation, we will proceed differently.
+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.
+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 $S$.
+Defining $\tilde C_2$ following the latter prescription, we find that, for the Simple Equation,
 \begin{equation}
-  \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))}
+  \begin{array}{>\displaystyle l}
+    \tilde C_2(\mathbf x)=
+    \rho^2\tilde g_2(\mathbf x)+
+    \\[0.3cm]
+    +\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))}
+  \end{array}
   \label{correlation_simpleq}
 \end{equation}
 where $\mathfrak K_{\tilde e}$ is the operator defined in\-~(\ref{Ke}).
-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 $|\mathbf x|^{-4}$.
-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.
-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.
-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 $|\mathbf x|^{-4}$.
-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.
+Defined in this way, $\tilde C_2\to\rho^2$ as $|\mathbf x|\to\infty$.
+\bigskip
+
+\indent
+$C_2$ 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}).
+The Big, Medium and Simple Equations make a natural prediction for the function $g_2$: namely $1-u(\mathbf x)$.
 \bigskip
 
 \indent
@@ -640,7 +647,7 @@ Note that this behavior was observed for the stronger potential $16e^{-|\mathbf
 \indent
 In Figure\-~\ref{fig:correlation}, we compare the prediction $\tilde C_2$ to the QMC simulation.
 At low densities, the prediction of the Big Equation agrees rather well with the QMC simulation.
-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}).
+The Simple and Medium Equations are not as accurate.
 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.
 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 $\tilde g_2$.
 At small $\mathbf x$, $\tilde C_2$ is negative, which is clearly not physical, and those values should be discarded.
-- 
cgit v1.2.3-54-g00ecf

