Glass transitions and scaling laws within an alternative mode-coupling theory

Idealized glass transitions are discussed within an alternative mode-coupling theory (TMCT) proposed by Tokuyama [Physica A 395, 31 (2014)]. This is done in order to identify common ground with and differences from the conventional mode-coupling theory (MCT). It is proven that both theories imply the same scaling laws for the transition dynamics, which are characterized by two power-law decay functions and two diverging power-law time scales. However, the values for the corresponding anomalous exponents calculated within both theories differ from each other. It is proven that the TMCT, contrary to the MCT, does not describe transitions with continuously vanishing arrested parts of the correlation functions. It is also demonstrated for a schematic model that the TMCT does not lead to the MCT scenarios either for transition-line crossings or for the appearance of higher-order glass-transition singularities.

Figure 1

Glass-transition diagram for a schematic model specified by a mode-coupling function $\mathcal{F}[P,x]=v_{1}x+v_3{x}^3$ within MCT [19] and TMCT. The dashed line marks the positions of continuous transition and the solid lines represent the positions of discontinuous transitions. Both circles mark the state point $({\upsilon }_1,{\upsilon }_3)$ with the maximum exponent parameter ${\lambda }_{\max }=2/3$ for TMCT (closed circle) and ${\lambda }_{\max }=1$ for MCT (open circle). The solid line for MCT terminates at the open circle.