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\inserttotalframenumber {20}} diff --git a/aps_mm_2017.pdf b/aps_mm_2017.pdf Binary files differindex 76e69ca..2dfbbea 100644 --- a/aps_mm_2017.pdf +++ b/aps_mm_2017.pdf diff --git a/aps_mm_2017.synctex.gz b/aps_mm_2017.synctex.gz Binary files differindex 7c38fdf..3282fe5 100644 --- a/aps_mm_2017.synctex.gz +++ b/aps_mm_2017.synctex.gz 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 Binary files differindex 9f3bbce..8264258 100644 --- a/figs/fig1.pdf +++ b/figs/fig1.pdf diff --git a/figs/fig11.pdf b/figs/fig11.pdf Binary files differindex d15e8c9..01fd1f5 100644 --- a/figs/fig11.pdf +++ b/figs/fig11.pdf diff --git a/figs/fig12.pdf b/figs/fig12.pdf Binary files differindex e519fad..21f6b0f 100644 --- a/figs/fig12.pdf +++ b/figs/fig12.pdf diff --git a/figs/fig13.pdf b/figs/fig13.pdf Binary files differindex 73cd23e..291c81c 100644 --- a/figs/fig13.pdf +++ b/figs/fig13.pdf diff --git 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