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 H(t) = integral(o)(l)Z(u) du and its inverse process T(t) = sup{u: H(u) less than or equal to t}. At the critical point T, these processes have ergodic limits when t --> infinity given by the stable additive process Y-a(t) and its inverse process X-a(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 a-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(-t(b)). (C) 2002 Elsevier Science B.V. All rights reserved.