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 F α ( q , t ) is first derived from the Newton equation with the memory function ψ α ( q , t ) , where α = c for the coherent–intermediate scattering function and s for the self–intermediate scattering function. In order to calculate ψ α ( q , t ) , the Mori type projection-operator method is then used and a linear non-Markov time-convolution equation for ψ α ( q , t ) is derived with the memory function φ α ( q , t ) . In order to calculate φ α ( q , t ) , the same binary approximation as that used in the mode-coupling theory (MCT) is also employed and hence φ α ( 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 f α ( q ) . In MCT it is given by f α ( q ) = Γ α ( q ) / ( Γ α ( q ) + 1 ) , where Γ α ( q ) is the long-time limit of the memory function φ α ( q , t ) . On the other hand, in the present theory it is given by f α ( q ) = exp [ − 1 / Γ α ( q ) ] . Thus, it is expected that the critical temperature T c of the present theory would be much lower than that of MCT. The other differences are also discussed.