From 31be95db9480de8f9b4c6e52f3f6845d2e231706 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 8 Dec 2020 14:22:14 +0100
Subject: Better formatting for partition function.
---
bezout.tex | 10 ++++++----
1 file changed, 6 insertions(+), 4 deletions(-)
diff --git a/bezout.tex b/bezout.tex
index da55794..bb35334 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -213,10 +213,12 @@ for $\delta=\kappa a^{-(p-2)}$.
Another instrument we have to study this problem is to compute the following partition function:
-\begin{eqnarray}
- Z(a,\beta)&=& \int \Pi_i dx_i dy_i \; e^{-\beta_{R} \Re H_0 -\beta_I \Im H_0}\nonumber\\
-& & \delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right)
-\end{eqnarray}
+\begin{equation}
+ \begin{aligned}
+ Z(a,\beta)&=\int dx\, dy \, e^{-\mathop{\mathrm{Re}}(\beta H_0)}\\
+ &\qquad\delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right)
+ \end{aligned}
+\end{equation}
The energy $\Re H_0, \Im H_0$ are in a one-to one relation with the temperatures $\beta_R,\beta_I$. The entropy $S(a,H_0) = \ln Z+ +\beta_{R} \langle \Re H_0 \rangle +\beta_I \langle \Im H_0\rangle$
is the logarithm of the number of configurations of a given $(a,H_0)$.
This problem may be solved exactly with replicas, {\em but it may also be simulated}
--
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