Significance of time-convolutionless mode-coupling theory in capturing the dynamics of glass-forming liquids

This paper demonstrates the significance of the recently proposed time-convolutionless mode-coupling theory (TMCT) in capturing the dynamics of glass-forming liquids. The origin of the primary differences between ideal MCT and TMCT is comprehensively explored from a unified perspective. First, we review two distinct projection operator methods in the Heisenberg picture: the time-convolution (TC) formulation proposed by Mori and the time-convolutionless (TCL) formulation proposed by Tokuyama and Mori. We show that the appropriate choice between these frameworks fundamentally depends on the space-time scales of the relevant variables. In TMCT, the TC formulation is applied to the current density, whereas the TCL formulation is applied to the number density because the latter operates on a significantly longer space-time scale. In contrast, ideal MCT applies the TC formulation to both densities. Consequently, the governing equation in TMCT is timeconvolutionless, whereas the ideal MCT equation features a time-convolution form. This fundamental difference significantly affects various physical quantities near the glass transition. Finally, the full TMCT equation is transformed into a simplified recursion equation to facilitate numerical analysis, and the key predictions of TMCT are compared with those of ideal MCT