From 5d0403945ad6bdeb111b1d7a25c03858e1443f4f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 8 Jun 2022 10:33:06 +0200
Subject: Removed unnecessary step in derivation of zero temperature free
 energy.

---
 frsb_kac_new.tex | 20 ++++----------------
 1 file changed, 4 insertions(+), 16 deletions(-)

(limited to 'frsb_kac_new.tex')

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)
-- 
cgit v1.2.3-70-g09d2

