Regenerative processes in supercooled liquids and glasses

L. Sjögren

Physica A: Statistical Mechanics and its Applications, 2003, vol. 322, issue C, 81-117

Abstract: 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)=∫0tZ(u)du 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.

Keywords: Mode-coupling theory; Regenerative processes; Selfsimilar processes; Regular variation; Ergodic limits (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:322:y:2003:i:c:p:81-117

DOI: 10.1016/S0378-4371(02)01832-0

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