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A simplified nonlinear memory function is proposed in the idealtime-convolutionless mode-coupling theory equation to study the dynamics ofglass-forming liquids. The numerical solutions are then compared with thesimulation results performed on fragile liquids and strong liquids. They areshown to recover the simulation results in a supercooled state well withinerror, except at a $\beta$-relaxation stage because of the ideal equation. Atemperature dependence of the nonlinearity $\mu$ in the memory function thensuggests that the supercooled state must be clearly separated into twosubstates, a weakly supercooled state in which $\mu$ increases rapidly as $T$decreases and a deeply supercooled state in which $\mu$ becomes constant up tothe glass transition as $T$ decreases. On the other hand, it is shown that in aglass state $\mu$ increases rapidly as $T$ decreases, while it is constant in aliquid state. Thus, it is emphasized that the new model for the simplifiedmemory function is much more reasonable than the conventional one proposedearlier by the present author not only qualitatively but also quantitatively.

A simplified nonlinear memory function for the dynamics of glass-forming materials based on time-convolutionless mode-coupling theory