From eb4ff41a5611b61caaffbb5c055df17791642ee7 Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Tue, 8 Dec 2020 10:58:58 +0000 Subject: Update on Overleaf. --- bezout.tex | 10 ++++++++-- phase.pdf | Bin 0 -> 6073 bytes 2 files changed, 8 insertions(+), 2 deletions(-) create mode 100644 phase.pdf diff --git a/bezout.tex b/bezout.tex index 3032076..62a773a 100644 --- a/bezout.tex +++ b/bezout.tex @@ -214,11 +214,17 @@ for $\delta=\kappa a^{-(p-2)}$. Another instrument we have to study this problem is to compute the following partition function: \begin{eqnarray} - Z= \int \Pi_i dx_i dy_i \; e^{-\beta_{R} \Re H_0 -\beta_I \Im H_0} - \delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right) + 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} 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} +Consider for example the ground-state energy for given $a$, that is, the energy in the limit $\beta_R \rightarrow \infty$ taken after $\beta_I \rightarrow \infty$. For $a=1$ this coincides with the ground-state of the real problem. + +\begin{center} + \includegraphics[width=4cm]{phase.pdf} +\end{center} } \bibliographystyle{apsrev4-2} diff --git a/phase.pdf b/phase.pdf new file mode 100644 index 0000000..2673d2f Binary files /dev/null and b/phase.pdf differ -- cgit v1.2.3-70-g09d2