Statistical-mechanical theory of nonlinear density fluctuations near the glass transition

The Tokuyama–Mori type projection-operator method is employed to study the dynamics of nonlinear density fluctuations near the glass transition. A linear non-Markov time-convolutionless equation for the scattering function Fα(q,t) is first derived from the Newton equation with the memory function ψα(q,t), where α=c for the coherent–intermediate scattering function and s for the self–intermediate scattering function. In order to calculate ψα(q,t), the Mori type projection-operator method is then used and a linear non-Markov time-convolution equation for ψα(q,t) is derived with the memory function φα(q,t). In order to calculate φα(q,t), the same binary approximation as that used in the mode-coupling theory (MCT) is also employed and hence φα(q,t) is shown to be identical with that obtained by MCT. Thus, the coupled equations are finally derived to calculate the scattering functions, which are quite different from the so-called ideal MCT equation. The most important difference between the present theory and MCT appears in the Debye–Waller factor fα(q). In MCT it is given by fα(q)=Γα(q)/(Γα(q)+1), where Γα(q) is the long-time limit of the memory function φα(q,t). On the other hand, in the present theory it is given by fα(q)=exp[−1/Γα(q)]. Thus, it is expected that the critical temperature Tc of the present theory would be much lower than that of MCT. The other differences are also discussed.