Time-convolutionless mode-coupling theory near the glass transition—A recursion formula and its asymptotic solutions

• Time-convolutionless mode coupling theory is analyzed mathematically and numerically.
• An ergodic to non-ergodic transition at a critical point is found.
• The same two-step relaxation process as that discussed in MCT is obtained.
• A recursion formula is solved for simulations numerically within a simplified model.

The time-convolutionless mode-coupling theory (TMCT) equation for the intermediate scattering function fα(q,t)fα(q,t) derived recently by the present author is analyzed mathematically and numerically, where α=cα=c stands for a collective case and α=sα=s for a self case. All the mathematical formulations discussed by Götze for the MCT equation are then shown to be directly applicable to the TMCT equation. Firstly, it is shown that similarly to MCT, there exists an ergodic to non-ergodic transition at a critical point, above which the long-time solution fα(q,t=∞)fα(q,t=∞), that is, the so-called Debye–Waller factor fα(q)fα(q), reduces to a non-zero value. The critical point is then shown to be definitely different from that of MCT. Secondly, it is also shown that there is a two-step relaxation process in a ββ stage near the critical point, which is described by the same two different power-law decays as those obtained in MCT. In order to discuss the asymptotic solutions, the TMCT equation is then transformed into a recursion formula for a cumulant function Kα(q,t)(=−ln[fα(q,t)]). By employing the same simplified model as that proposed by MCT, the simplified asymptotic recursion formula is then numerically solved for different temperatures under the initial conditions obtained from the simulations. Thus, it is discussed how the TMCT equation can describe the simulation results within the simplified model.