Diffusion Scaling in Event-Driven Random Walks: An Application to Turbulence

Abstract

Scaling laws for the diffusion generated by three different random walk models are reviewed. The random walks, defined on a one-dimensional lattice, are driven by renewal intermittent events with non-Poisson statistics and inverse power-law tail in the distribution of the inter-event or waiting times, so that the event sequences are characterized by self-similarity. Intermittency is a ubiquitous phenomenon in many complex systems and the power exponent of the waiting time distribution, denoted as complexity index, is a crucial parameter characterizing the system's complexity. It is shown that different scaling exponents emerge from the different random walks, even if the self-similarity, i.e. the complexity index, of the underlying event sequence remains the same. The direct evaluation of the complexity index from the time distribution is affected by the presence of added noise and secondary or spurious events. It is possible to minimize the effect of spurious events by exploiting the scaling relationships of the random walk models. This allows to get a reliable estimation of the complexity index and, at the same time, a confirmation of the renewal assumption. An application to turbulence data is shown to explain the basic ideas of this approach.