On the relation between a length cutoff in time-convolutionless mode-coupling theory and a characteristic length at β-relaxation stage in glass-forming materials

A length cutoff b contained in the nonlinear memory function of the time-convolutionless mode-coupling theory (TMCT) equation is obtained by solving the TMCT equation in a manner consistent with the simulation results near the glass transition. A characteristic length ℓ of a supercooled liquid is also introduced at a β-relaxation stage based on the mean-field theory proposed by Tokuyama independently and is shown to describe a displacement of a particle in a cage. Then, both lengths are shown to satisfy the inequality ℓ≥b≥bc in a supercooled state within an original TMCT equation, where bc is a critical cutoff obtained independently by solving the Lambert W-function at the critical point. Their control parameter dependence is also explored from a unified point of view. Thus, both lengths are shown to characterize the same caging mechanism at β stage in a supercooled liquid.