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Diffstat (limited to '2-point.tex')
| -rw-r--r-- | 2-point.tex | 27 |
1 files changed, 25 insertions, 2 deletions
diff --git a/2-point.tex b/2-point.tex index 6e208cd..b187e96 100644 --- a/2-point.tex +++ b/2-point.tex @@ -87,9 +87,9 @@ stationary point of energy density $E_1$ and stability $\mu_1$. \end{align*} \begin{align*} \Sigma_{12} - &=\frac1N\lim_{n\to0}\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\, + &=\frac1N\lim_{n\to0}\lim_{m\to-1}\overline{\int d\nu_H(\mathbf s_0\mid E_0,\mu_0)\left(\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)\right)^m\, \frac\partial{\partial n}\bigg(\int d\nu_H(\mathbf s_1\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_1)\bigg)^n}\\ - &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\int\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_a)} + &=\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\pmb \sigma_1\cdot \mathbf s_a)\right)} \end{align*} \begin{equation} \overline{\big|\det\operatorname{Hess}H(s)\big|\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(s)\big)} @@ -118,6 +118,29 @@ stationary point of energy density $E_1$ and stability $\mu_1$. \begin{align*} &\Sigma_{12} + =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int e^{Nm\mathcal S_0(\hat\beta_0,C^{00},R^{00},D^{00})+Nn\mathcal S_1(\hat\beta_0,\hat\beta_1,C^{00},C^{01},C^{11},R^{00},R^{01},R^{10},R^{11},D^{00},D^{01},D^{11})} +\end{align*} + +\begin{align} + C^{01} + =\begin{bmatrix} + q&\cdots&q\\ + q'&\cdots&q'\\ + \vdots&\ddots&\vdots\\ + q'&\cdots&q' + \end{bmatrix} + && + R^{01} + =\begin{bmatrix} + r_{01}&\cdots&r_{01}\\ + r_{01}'&\cdots&r_{01}'\\ + \vdots&\ddots&\vdots\\ + r_{01}'&\cdots&r_{01}' + \end{bmatrix} +\end{align} + +\begin{align*} + &\Sigma_{12} =\frac1N\frac{e^{-\hat\beta_0E_0-r_0\mu_0+\frac12\left[\hat\beta_0^2f(1)-(2\hat\beta_0r_0^2+d_0)f'(1)+r_0^2f''(1)\right]+\mathcal D(\mu_0)}}{e^{N\Sigma(E_0,\mu_0)}}+\mathcal D(\mu_1)+\hat\beta_1E_1-\frac12\hat\mu_1-\mu_0r_{00} +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r_{01}+\hat\beta_1r_{10}-d_{01})f'(q)+r_{01}r_{10}f''(q) \\& |