From e40fc5e3e906682bd3764d8584550a480c4728a9 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 16 Sep 2019 16:03:19 -0400
Subject: coarse removal of ginzburg, needs more smoothing

---
 main.tex | 45 ++++-----------------------------------------
 1 file changed, 4 insertions(+), 41 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 3928eae..d7cd7cf 100644
--- a/main.tex
+++ b/main.tex
@@ -479,47 +479,10 @@ This is typically done by computing the Ginzburg temperature
 \cite{ginzburg_remarks_1961}, which gives the proximity to the critical point
 $t=(T-T_c)/T_c$ at which mean field theory is expected to break down by
 comparing the magnitude of fluctuations in a correlation-length sized box to
-the magnitude of the field. In the modulated phase the spatially averaged
-magnitude is zero, and so we will instead compare fluctuations in the
-\emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the
-field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the
-modulated phase $\langle\alpha(x)\rangle=\alpha_0$ for $\alpha_0^2=|\delta
-\tilde r|/4u$. In the modulated phase, the $q$-dependant fluctuations in
-$\alpha$ are given by
-\begin{equation}
-  G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|},
-\end{equation}
-An estimate of the Ginzburg criterion is then given by the temperature at which
-$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle_{V_\xi}\simeq\langle\alpha\rangle^2=\alpha_0^2$,
-where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$
-can be replaced in favor of the jump in the specific heat at the transition
-using
-\begin{equation}
-  c_V=-T\frac{\partial^2f}{\partial T^2}
-    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases}
-\end{equation}
-The integral over the correlation function $G_\alpha$ can be preformed up to
-one integral analytically using a Gaussian-bounded correlation volume, yielding
-$\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}|\Delta \tilde r|^{-1}\mathcal
-I(\xi_\perp q_*)$ for
-\begin{equation}
-  \begin{aligned}
-    \mathcal I(x)=-2^{3/2}\pi^{5/2}\int& dy\,e^{[2+(4x^2-1)y^2+y^4]/2} \\
-      &\times\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big).
-  \end{aligned}
-\end{equation}
-This gives a transcendental equation
-\begin{equation}
-  \mathcal I(\xi_{\perp0}q_*t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}t_\G^{3/4},
-\end{equation}
-with $\xi=\xi_0|t|^{-\nu}$ defining the bare correlation lengths.
-Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
-\cite{fisher_specific_1990}. Our fit above gives $\xi_{\perp0}q_*=(D_\perp
-q_*^4/aT_c)^{1/4}\simeq2$. Further supposing that
-$\xi_{\parallel0}\simeq\xi_{\perp0}$, we find $t_\G\sim0.4$, though this
-estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our
-estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in
-$t_\G$ over nearly four orders of magnitude.  The estimate here predicts
+the magnitude of the field.
+
+Our fit above gives $\xi_{\perp0}q_*=(D_\perp
+q_*^4/aT_c)^{1/4}\simeq2$. The estimate here predicts
 that an experiment may begin to see deviations from mean field behavior within
 around several degrees Kelvin of the critical point. A \rus\ experiment with more precise
 temperature resolution near the critical point may be able to resolve a
-- 
cgit v1.2.3-70-g09d2

