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| -rw-r--r-- | frsb_kac_new.tex | 20 |
1 files changed, 4 insertions, 16 deletions
diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex index e1c5768..beeab96 100644 --- a/frsb_kac_new.tex +++ b/frsb_kac_new.tex @@ -143,22 +143,10 @@ $q_0=0$ \right] \right) \end{align*} -The zero temperature limit is most easily obtained by putting $x_i=\tilde x_ix_k$ and $x_k=y/\beta$, $q_k=1-z/\beta$ -\begin{align*} - \beta F= - -\frac12\log S_\infty- - \frac12\left(\beta^2f(1)+\beta^2(y\beta^{-1}-1)f(1-z\beta^{-1})+y\beta\sum_{i=0}^{k-1}(\tilde x_i-\tilde x_{i+1})f(q_i)\right. \\ - +\frac\beta{\tilde x_1 y}\log\left[ - y\sum_{i=1}^{k-1}(\tilde x_i-\tilde x_{i+1})q_i+y+z-yz/\beta - \right]\\ - +\sum_{j=1}^{k-1}\frac\beta y(\tilde x_{j+1}^{-1}-\tilde x_j^{-1})\log\left[ - y\sum_{i=j+1}^{k-1}(\tilde x_i-\tilde x_{i+1})q_i+y+z-yz/\beta-y\tilde x_{j+1}q_j -\right]\\ - \left.-\frac\beta{\tilde x_1 y}\log\beta-\sum_{j=1}^{k-1}\frac\beta y(\tilde x_{j+1}^{-1}-\tilde x_j^{-1})\log\beta+(1-\beta y^{-1})\log\left[ - z/\beta - \right] -\right) -\end{align*}Taking the limit we get +The zero temperature limit is most easily obtained by putting $x_i=\tilde +x_ix_k$ and $x_k=y/\beta$, $q_k=1-z/\beta$, which ensures the $\tilde x_i$, +$y$, and $z$ have nontrivial limits. Inserting the ansatz and taking the limit +we get \begin{align*} \lim_{\beta\to\infty}F= -\frac12\left(yf(1)+zf'(1)+y\sum_{i=0}^{k-1}(\tilde x_i-\tilde x_{i+1})f(q_i) |