1208.1383 (Michael A. Lomholt et al.)

Non-renewal counting processes in systems with ageing waiting times:
microscopic origin of logarithmic time evolution    [PDF]

Michael A. Lomholt, Tobias Ambjornsson, Ludvig Lizana, Ralf Metzler

We study a generic counting process for systems governed by ageing waiting times. In contrast to renewal continuous time random walks with independent waiting times, we consider the case when each state is characterized by its own internal waiting time process, having been initiated at time t=0. Therefore each transition from state n to n+1 is triggered by psi_1, the forward waiting time density, instead of the regular waiting time density psi. For states characterized by heavy-tailed forms of psi we obtain an asymptotically logarithmic evolution of the counting dynamics n(t), whose fluctuations vanish relatively to the mean. The counting process introduced here describes the dynamics of crack propagation, the counting of on-off transitions in ensembles of blinking quantum dots, or, more generally, the dynamics in complex systems such as glasses or biological cells.

View original: http://arxiv.org/abs/1208.1383