summaryrefslogtreecommitdiff
diff options
context:
space:
mode:
authorJaron Kent-Dobias <jaron@kent-dobias.com>2022-01-07 16:53:23 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2022-01-07 16:53:23 +0100
commit66bd034adc5bbb782e06780b8f1625ec0a53332e (patch)
tree6265069249f20be4a2053691eb6b3fb7cd4506de
parent6539131ba2e5652f6cb70f83a9144eb0b99254a6 (diff)
downloadJPA_55_434006-66bd034adc5bbb782e06780b8f1625ec0a53332e.tar.gz
JPA_55_434006-66bd034adc5bbb782e06780b8f1625ec0a53332e.tar.bz2
JPA_55_434006-66bd034adc5bbb782e06780b8f1625ec0a53332e.zip
Started some notes about the 2-spin partition function.overleaf
-rw-r--r--stokes.tex14
1 files changed, 14 insertions, 0 deletions
diff --git a/stokes.tex b/stokes.tex
index 91f7f18..70a7ed7 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -392,6 +392,20 @@ separatrix of a third. This means that when the imaginary energies of two
critical points are brought to the same value, their surfaces of constant
imaginary energy join.
+\begin{equation}
+ \begin{aligned}
+ Z(\beta)
+ &=\int_{S^{N-1}}dx\,e^{-\beta H(x)}
+ =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\
+ &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\
+ &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\
+ &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\det(\beta J+\lambda I)}}e^{\lambda N} \\
+ &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\prod_i(\beta\lambda_i+\lambda)}}e^{\lambda N} \\
+ &=(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac12\sum_i\log(\beta\lambda_i+\lambda)} \\
+ &\simeq(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac N2\int d\lambda'\,\rho(\lambda')\log(\beta\lambda'+\lambda)} \\
+ \end{aligned}
+\end{equation}
+
\subsection{Pure \textit{p}-spin}
\begin{equation}