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Diffstat (limited to 'main.tex')
| -rw-r--r-- | main.tex | 12 |
1 files changed, 6 insertions, 6 deletions
@@ -181,7 +181,7 @@ which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cann \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1} &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')} =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\ - &=-\frac2b\chi^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta(x-x') + &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta_{ij}\delta(x-x') \end{aligned} \label{eq:inv.func} \end{equation} @@ -191,25 +191,25 @@ It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the suscept \begin{aligned} \chi_{\X ij}^{-1}(x,x') &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ - &=\lambda_\X\delta(x-x')+ + &=\lambda_\X\delta_{ij}\delta(x-x')+ b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')} +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\ &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ - &=\lambda_\X\delta(x-x')+ + &=\lambda_\X\delta_{ij}\delta(x-x')+ b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')} -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ - &=\lambda_\X\delta(x-x')+ + &=\lambda_\X\delta_{ij}\delta(x-x')+ b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')} -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} - =\lambda_\X\delta(x-x')+ + =\lambda_\X\delta_{ij}\delta(x-x')+ \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}, \end{aligned} \end{equation} \end{widetext} whose Fourier transform follows from \eqref{eq:inv.func} as \begin{equation} - \chi_{\X ij}(q)=\frac1{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q). + \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q). \label{eq:elastic.susceptibility} \end{equation} At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$. |