From 19398b20a690a17fc82fcf9ad4c0f87f4de9f174 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 11 Mar 2017 19:45:29 -0500
Subject: progress

---
 aps_mm_2017.tex | 56 +++++++++++++++++++++++++++++++++++++++++---------------
 1 file changed, 41 insertions(+), 15 deletions(-)

(limited to 'aps_mm_2017.tex')

diff --git a/aps_mm_2017.tex b/aps_mm_2017.tex
index 89f54ed..2bcd785 100644
--- a/aps_mm_2017.tex
+++ b/aps_mm_2017.tex
@@ -136,7 +136,8 @@
   \[
     \Delta F=\Sigma N^\sigma-MHN
   \]
-  \pause
+  with $1-\frac1d\leq\sigma<1$.
+  \pause\vspace{1em}\\
   Metastable phase is stable to domains smaller than
   \[
     N_\crit=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
@@ -145,27 +146,52 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{The Metastable Ising Model}
-
-  The formation of a critical domain has energy cost
+  \frametitle{Metastability \& Complex Free Energy}
+  Formation of critical domain has energy cost
   \[
-    \Delta F_\crit\sim MH\bigg(\frac{MH}{\Sigma}\bigg)^{-1/(1-\sigma)}
+    \Delta F_\crit=\Delta F\Big|_{N=N_\crit}\sim
+    \bigg(\frac{\Sigma}{(MH)^\sigma}\bigg)^{1/(1-\sigma)}
   \]
-
   \pause
-
-  The decay rate of the metastable is proportional to the probability of
-  forming a critical domain $e^{-\beta\Delta F_\crit}$.
-  
-  \pause \vspace{1em}
-  
-  Decay of the equilibrium state implies existence of an imaginary part in the
-  free energy,
+  Probability of such a domain forming is
+  \[
+    P_\crit\sim e^{-\beta\Delta F_\crit}
+  \]
+  \pause
+  Imaginary free energy is therefore
   \[
-    \im F\sim e^{-\beta\Delta F_\crit}
+    \im F\sim\Gamma\sim P_\crit\sim e^{-\beta\Delta F_\crit}
+    =e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}}
   \]
 
 \end{frame}
+
+\begin{frame}
+  \frametitle{Metastability \& Complex Free Energy}
+  \begin{columns}
+    \begin{column}{0.4\textwidth}
+      \includegraphics{figs/fig1}
+    \end{column}
+    \begin{column}{0.6\textwidth}
+      $\im F$ has an essential singularity of the form
+      $e^{-1/H^{\sigma/(1-\sigma)}}$.
+      \vspace{1em}\pause\\
+      Near critical point, $\sigma=1-\frac1d$, and
+      \[
+        \im F\sim e^{-1/H^{d-1}}
+      \]
+      \pause
+      Nonanalytic behavior is universal!
+      \vspace{1em}\pause\\
+      Can directly observe by measuring metastable decay rate, but what else?
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Analytic Constraints on the Stable Free Energy}
+\end{frame}
+
 \begin{frame}
   \frametitle{The Metastable Ising Model}
 
-- 
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