Aging renewal theory and application to random walks

By Johannes H. P. Schulz, Eli Barkai, Ralf Metzler. 2013

140 citations recorded by the source corpus.

The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. We here discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long time properties, this process exhibits aging: events counted initially in a time interval [0,t] statistically strongly differ from those observed at later times [t_a,t_a+t]. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous time random walk. By this we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking. Along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.

Subfields: Condensed Matter Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8a49f39b2e039877ad203164c68c849dc9fa30de

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In Condensed Matter Physics

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Citations in the corpus

Protein Crowding in Lipid Bilayers Gives Rise to Non-Gaussian Anomalous Lateral Diffusion of Phospholipids and Proteins (2016, background): “exhibits pronounced aging effects [55,58,63], a phenomenon not observed in our much simpler model membranes.”
Effects of temporal correlations in social multiplex networks (2016, background): “Analytic expressions for this function can be obtained in Laplace space[7].”
L\'evy walks (2014, background): “In addition, understanding of the fluctuations in time averaged observables can help to obtain more information about the underlying stochastic process (Dechant et al., 2014; Schulz et al., 2014).”