Abstract
We 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 ½0;tstatistically strongly differ from those observed at later times ½ta;ta þ t� . 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. 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.
Highlights
A stochastic process nðtÞ counting the number of some sort of events occurring during a time interval 1⁄20; t is called a renewal process, if the time spans between consecutive events are independent, identically distributed random variables [1]
Maybe the most obvious physical application is the counting of decays from a radioactive substance. This is an example of a Poissonian renewal process: the random time passing between consecutive decay events, the waiting time, has an exponential probability density function ψðtÞ 1⁄4 τ−1 expð−t=τÞ
We demonstrate scaling convergence: if a renewal process with simple power-law waiting time distribution is monitored on increasingly long scales for time and event numbers, its statistics approach the continuous limit described by Eqs. (11–13)
Summary
A stochastic process nðtÞ counting the number of some sort of events occurring during a time interval 1⁄20; t is called a renewal process, if the time spans between consecutive events are independent, identically distributed random variables [1]. Divergent Renewal processes of this type are said to be scale free, since, roughly speaking, statistically dominant waiting times are always of the order of the observational time Their outstanding characteristics play out most severely on long time scales: while for t ≫ τ, Poissonian renewal processes behave quasideterministically, nðtÞ ≈ t=τ, heavy-tailed distributions lead to nontrivial random properties at all times. Stochastic processes of this type are known to exhibit weak ergodicity breaking [12]; i.e., time averages and associated ensemble averages of a physical observable are not equivalent. We discuss the intricate interplay of aging and relaxation modes and highlight the essential features and pitfalls for aged ensemble and singletrajectory measurements
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