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Diffstat (limited to 'bezout.tex')
| -rw-r--r-- | bezout.tex | 10 |
1 files changed, 8 insertions, 2 deletions
@@ -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} |