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| -rw-r--r-- | bezout.tex | 10 |
1 files changed, 6 insertions, 4 deletions
@@ -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} |