Aging Renewal Theory and Application to Random Walks

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 [0,t] statistically strongly differ from those observed at later times [ta,ta+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.

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Plain Language Summary

Ticks on the financial market, people entering a concert hall, or microscopic particles diffusing across an osmotic membrane are all examples of stochastic events. If such events occurred independently from each other, the underlying processes would all correspond to so-called renewal processes. In the simplest case, the waiting times between two successive events would follow the Poisson distribution—well known from the decay of a radioactive material. In many cases, however, we encounter anomalous renewal processes with pronounced non-Poissonian distributions. Types of anomalous renewal processes that are particularly interesting are characterized by a power-law waiting-time distribution. Indeed, such processes show “aging”—the behavior of the system changes forever during its evolution—which is observed in a broad range of phenomena from the blinking of quantum dots to the motion of submicron particles in living biological cells. In this theoretical paper, we study these processes and answer the question of what effects a particular period of aging has on the process dynamics that follows it.

Generally, a renewal process corresponds to the number of transitions in a system—the ticks of a Geiger counter or the jumps of a random walker. For the strongly disordered systems we have in mind, we show that the older the system in question, the slower its renewal process: The on-off blinking of quantum dots becomes less frequent, or the motion of a protein channel in a cell wall increasingly stalls. In a very old system, hardly any events occur. We present an integrated approach to the aging renewal theory with a careful discussion of hitherto neglected phenomena. Thus, we detail how, in an ensemble of particles, the population splits into a discrete fraction that does not show any renewals at all during a certain measurement period, and a fraction with a continuous distribution of renewals. Moreover, we show that aging effects show up differently in ensemble and time averages of the same physical observable and also that a physical interpretation is more straightforward using time averages of aging systems. Without knowledge of this subtle effect, one risks misinterpreting experimental or simulations data obtained by different measurements of the same process.

Our theory can be applied to a wide range of systems, including those mentioned above. Because of its generic character, we also expect it to prove useful in finance and queuing theory, or in a wider context such as operations research, risk modeling, and social processes.