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| -rw-r--r-- | ising_scaling.tex | 10 |
1 files changed, 5 insertions, 5 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index df4f15b..c25dfe2 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -208,7 +208,7 @@ sufficiently large equilibrium `bubble' occurs. The bulk of such a bubble of radius $R$ lowers the free energy by $2M|H|\pi R^2$, where $M$ is the magnetization, but its surface raises the free energy by $2\pi R\sigma$, where $\sigma$ is the surface tension between the -stable--metastable interface. The bubble is sufficiently large to decay +stable--metastable interface. The bubble is sufficiently large to catalyze the decay of the metastable state when the differential bulk savings outweigh the surface costs. This critical bubble occurs with free energy cost \begin{equation} @@ -225,7 +225,7 @@ metastable phase, with \begin{equation} \operatorname{Im}F\propto\Gamma\sim e^{-\beta\Delta F_c}\simeq e^{-1/b|\xi|} \end{equation} -which can be more rigorously related in the context of quantum field theory +which can be more rigorously derived in the context of quantum field theory \cite{Voloshin_1985_Decay}. The constant $b=2M_0/\pi\sigma_0^2$ is predicted by known properties, e.g., for the square lattice $M_0$ and $\sigma_0$ are both predicted by Onsager's solution \cite{Onsager_1944_Crystal}, but for our @@ -245,8 +245,8 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \end{figure} To lowest order, this singularity is a function of the scaling invariant $\xi$ -alone. It is therefore suggestive that this should be considered a part of the -singular free energy and moreover part of the scaling function that composes +alone. This suggests that it should be considered a part of the +singular free energy, and thus part of the scaling function that composes it. There is substantial numeric evidence for this as well \cite{Enting_1980_An, Fonseca_2003_Ising}. We will therefore make the ansatz that @@ -373,7 +373,7 @@ entirely fixed, and it will be truncated at finite order. \caption{ Example of the parametric coordinates. Lines are of constant $R$ from $-\theta_0$ to $\theta_0$ for $g(\theta)$ taken from the $n=6$ entry of - Table \ref{tab:fits}. + Table \ref{tab:fits}. {\color{blue} \bf XXX Can we have lines of constant $\theta$ as well? Maybe dashed? Also maybe smaller radii, $R=1/4$, 1/2, and 1? Legend could be } \label{fig:schofield} \end{figure} |