5 . Waiting - Time Distributions

Somewhere, something incredible is waiting to be known. The universe is full of magical things, patiently waiting for our wits to grow sharper. Why are waiting times interesting? While waiting times in our everyday life are usually associated with boredom, the statistics of waiting times contains scientifically interesting information about (1) the mean rates of event occurrence and (2) the randomness and un-correlatedness of events, in contrast to events that are causally related in a system with local correlations or long-range interactions. Especially in astrophysical observations, event statistics of stellar flares or gamma-ray bursts from black-hole candidates are gathered without any spatial information, and thus we cannot decide whether individual events originate from the same source, or from spatially independent sources, which can only be discriminated by means of waiting-time statistics. Similar issues arise for solar flares and magnetospheric substorms, where spatial localization has often limited accuracy and mutual interactions are hidden by the invisible magnetic field. The classical SOC model of sandpile avalanches predicts no correlation between the input (of dripping sand grains) and the output (of sand avalanches) in the state of self-organized criticality, which implies that the statistics of waiting times (between subsequent avalanches) should be strictly random and consistent with Poisson statistics (e.g., for numerical simulations of cellular automa-tons see, e.g., waiting-time distribution in Fig. 6 of Charbonneau et al. 2001). There are physical processes that produce similar powerlaw distributions as SOC processes do, such as intermittent turbulence, but the two processes can be discriminated from their different waiting-time distributions, because SOC processes are uncorrelated, while turbulence entails spatial correlations. Waiting times are also called elapsed times, inter-occurrence times, or laminar times, in analogy to hydrodynamic flows in a laminar or turbulent state. Earthquakes were found to have a different waiting-time statistics than those of after-shocks, because earthquakes from different regions and continents are uncorrelated, while