Regenerative processes in supercooled liquids and glasses

The mode-coupling equations used to study glasses and supercooled liquids define the underlying regenerative processes represented by an indicator function Z(t). Such a process is a special case of an alternating renewal process, and it introduces in a natural way a stochastic two level system. In terms of the fundamental Z-process one can define several other processes, such as a local time process and its inverse process T(t)=sup{u : H(u) t}. At the critical point Tc these processes have ergodic limits when t→∞ given by the stable additive process Ya(t) and its inverse process Xa(t), where a is the critical exponent. These processes are selfsimilar, and the latter is given by the Mittag-Leffler distribution. The appearance of these limit processes, which is a consequence of the Darling–Kac theorem, is the generic reason for the universal predictions of the mode-coupling theory, and are observed in many glassforming systems. We also find a similar behaviour for the α-relaxation function but for the initial behaviour at t→0, and the limit processes are in this case given by Y1−b and X1−b, where b is the von Schweidler exponent. This also implies that the relaxation function belongs to the domain of attraction of the stable distribution with the characteristic function exp(−tb).