Recursion formulas for nonlinear density fluctuations near the glass transition

Abstract

The time-convolutionless mode-coupling (TMCT) equation for the intermediate scattering function $f_{\alpha }(q,t)$ derived recently by the present author is transformed into a simple nonlinear recursion formula for a generating function ${\Omega }_{\alpha }(\mathbf{q},t)(=-\ln [f_{\alpha }(q,t)]/q^2)$, where $\alpha =c$ stands for a collective case and $\alpha =s$ for a self case. By employing the same simplification on the nonlinear memory function as that proposed by the mode-coupling theory (MCT), the simplified asymptotic recursion formula is then derived and is numerically analyzed for different temperatures under the initial conditions obtained from the simulation. In a liquid state the numerical results are shown to recover the simulation results well. Although they can describe the simulation results well in the $\beta$-relaxation stage even for lower temperatures, they do not agree with those in the so-called $\alpha$-relaxation stage because of the simplified model. The coupling parameter ${\lambda }^{(\alpha )}$ dependence of the Debye-Waller factor $f_{\alpha }$ is also discussed. The critical point is found as ${\lambda }_c^{(c)}=2e(\simeq 5.43656)$ and $f_c=e^{-1/2}(\simeq 0.60653)$, while MCT gives ${\lambda }_c^{(c)}=4.0$ and $f_c=1/2$. Then, the critical temperature $T_c$ is shown to be definitely lower than that predicted by MCT. Thus, it is emphasized that the present theory can improve the high $T_c$ problem appeared in MCT. The time evolution of the memory function and that of the diffusion coefficient are also investigated within asymptotic formulas.

arXiv commentThis paper has been withdrawn by the author due to a crucial mistake in the asymptotic recursion formula given by Eq. (40)

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