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Regenerative processes in supercooled liquids and glasses

Journal article
Authors Lennart Sjögren
Published in Physica A
Volume 322
Pages 81-117
ISSN 0378-4371
Publication year 2003
Published at Department of Physics (GU)
Pages 81-117
Language en
Keywords Mode-coupling theory; Regenerative processes; Selfsimilar processes; Regular variation; Ergodic limits
Subject categories Physical Sciences, Statistical physics


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 Image and its inverse process T(t)=sup{u : H(u)less-than-or-equals, slantt}. 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).

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