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authorJaron Kent-Dobias <jaron@kent-dobias.com>2017-03-11 19:45:29 -0500
committerJaron Kent-Dobias <jaron@kent-dobias.com>2017-03-11 19:45:29 -0500
commit19398b20a690a17fc82fcf9ad4c0f87f4de9f174 (patch)
tree4484117f208e44ddc133fcd12882b8483310ebad
parent828ae03eb71aa24b0a1fda13f02684a3df3202eb (diff)
downloadaps_mm_2017-19398b20a690a17fc82fcf9ad4c0f87f4de9f174.tar.gz
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diff --git a/aps_mm_2017.pdf b/aps_mm_2017.pdf
index 76e69ca..2dfbbea 100644
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+++ b/aps_mm_2017.pdf
Binary files differ
diff --git a/aps_mm_2017.synctex.gz b/aps_mm_2017.synctex.gz
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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}
diff --git a/figs/fig1.pdf b/figs/fig1.pdf
index 9f3bbce..8264258 100644
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diff --git a/figs/fig11.pdf b/figs/fig11.pdf
index d15e8c9..01fd1f5 100644
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diff --git a/figs/fig12.pdf b/figs/fig12.pdf
index e519fad..21f6b0f 100644
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Binary files differ
diff --git a/figs/fig13.pdf b/figs/fig13.pdf
index 73cd23e..291c81c 100644
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diff --git a/figs/fig14.pdf b/figs/fig14.pdf
index b0619bf..eda9a41 100644
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diff --git a/figs/fig4.pdf b/figs/fig4.pdf
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