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| -rw-r--r-- | essential-ising.tex | 48 |
1 files changed, 23 insertions, 25 deletions
diff --git a/essential-ising.tex b/essential-ising.tex index 0fb23a0..0b631e5 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -36,9 +36,6 @@ \maketitle -\begin{abstract} -\end{abstract} - It's long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by \cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic} @@ -82,23 +79,27 @@ For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes \[ \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}} \] -Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ -\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$ -with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$, +There are known scaling forms for the surface tension and magnetization, $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$ +\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$. +Since both the surface tension and magnetization have nonzero finite values at +the first-order transition $h=0$, $\mathcal S(0)=\O(1)$ and $\mathcal +M(0)=\O(1)$. It follows that \[ \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}} \] with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$ -besides the prefactor. Results from field theory predict that, for small $H$ -and $1<d<5$, $d\neq 3$, -\[ - \im F\simeq\bigg(\frac h{t^\Delta}\bigg)^{-(d-3)d/2}(g^*)^{-d(d-1)/4} - \exp\bigg[-B\bigg(\frac h{|t|^\Delta}\bigg)^{-(d-1)}(g^*)^{-(d+1)/2}\bigg] -\] -\[ - \im F\simeq\bigg(\frac - h{t^\Delta}\bigg)^{-7/3}(g^*)^{-8/3}\exp\bigg[-B\bigg(\frac - h{t^\Delta}\bigg)^{-2}(g^*)^{-2}\bigg] +besides the prefactor. Results from field theory predict that, for small $h$, +\[ + \im F\simeq + \begin{cases} + \big(\frac + h{t^\Delta}\big)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}\exp\big[-B\big(\frac + h{|t|^\Delta}\big)^{-(d-1)}(g^*)^{-(d+1)/2}\big] & d=2,4\\ + \big(\frac + h{t^\Delta}\big)^{-7/3}(g^*)^{-8/3}\exp\big[-B\big(\frac + h{t^\Delta}\big)^{-2}(g^*)^{-2}\big] + & d=3 + \end{cases} \] with $\Delta=3-\frac\epsilon2$, $g^*=2\pi^2\frac\epsilon{n+8}$ \cite{houghton.1980.metastable,gunther.1980.goldstone}. This is consistent @@ -116,15 +117,12 @@ with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|} \] -\cite{langer.1967.condensation} - -\[ - F(X)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X' - =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd - X' - =-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd - X' -\] +\begin{align} + F(X) + &=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X' + =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd X'\\ + &=-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd X' +\end{align} since $\im F=0$ for $X>0$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$. Unfortunately this integral doesn't converge, and it seems we cannot evaluate this result at the level of truncation we've chosen. However, |