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Diffstat (limited to '2-point.tex')
| -rw-r--r-- | 2-point.tex | 35 |
1 files changed, 20 insertions, 15 deletions
diff --git a/2-point.tex b/2-point.tex index bf1a5ca..efc7a71 100644 --- a/2-point.tex +++ b/2-point.tex @@ -670,28 +670,33 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) && \tilde d_\mathrm d^{00}=d^{00}_\mathrm df'(1) \end{align} - - +Once these expressions are inserted into the complexity, the limits of $n$ and +$m$ to zero can be taken, and the parameters from $D^{01}$ and $D^{11}$ can be +extremized explicitly. The resulting expression for the complexity, which must +still be extremized over the parameters $\hat\beta_1$, $r^{01}$, +$r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is \begin{equation} \begin{aligned} - &\Sigma_{12}=\mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1 - +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big) - +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q) - \\ - &+\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)+ - \frac12\Bigg\{ + &\Sigma_{12}(E_0,\mu_0,E_1,\mu_1,q) + =\mathop{\mathrm{extremum}}_{\hat\beta_1,r^{11}_\mathrm d,r^{11}_0,r^{01},q^{11}_0}\Bigg\{ + \mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1 + +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)\\ + &\qquad+\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q) + +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)\\ + &+\frac12\Bigg[ \hat\beta_1^2\big(f(1)-f(q^{11}_{0})\big) +(r^{11}_\mathrm d)^2f''(1)+2r^{01}r^{10}f''(q)-(r^{11}_0)^2f''(q^{11}_0) + +\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1) \\ - &+\left( - (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right) + &\qquad+\frac{1-q^2}{1-q^{11}_0}+\left( + (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0} + \left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right) \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \\ - &+\frac{1-q^2}{1-q^{11}_0}+\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1) + &\qquad -\frac1{f'(1)}\frac{f'(1)^2-f'(q)^2}{f'(1)-f'(q^{11}_0)} +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big) - \\ - &+\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right) - \Bigg\} + +\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right) + \Bigg]\Bigg\} \end{aligned} \end{equation} @@ -713,7 +718,7 @@ expanding in powers of $\Delta q=1-q$. For the complexity, the result is \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q) +O\big((1-q)^2\big) \end{equation} -The popular of stationary points that are most common at each energy have the relation +The population of stationary points that are most common at each energy have the relation \begin{equation} E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0 \end{equation} |