Diffusion entropy and waiting time statistics of hard x-ray solar flares

Abstract

We analyze the waiting time distribution of time distances $\tau$ between two nearest-neighbor flares. This analysis is based on the joint use of two distinct techniques. The first is the direct evaluation of the distribution function $\psi(\tau)$, or of the probability, $\Psi(tau)$, that no time distance smaller than a given $\tau$ is found. We adopt the paradigm of the inverse power law behavior, and we focus on the determination of the inverse power index $\mu$, without ruling out different asymptotic properties that might be revealed, at larger scales, with the help of richer statistics. The second technique, called Diffusion Entropy (DE) method, rests on the evaluation of the entropy of the diffusion process generated by the time series. The details of the diffusion process depend on three different walking rules, which determine the form and the time duration of the transition to the scaling regime, as well as the scaling parameter $\delta$. With the first two rules the information contained in the time series is transmitted, to a great extent, to the transition, as well as to the scaling regime. The same information is essentially conveyed, by using the third rules, into the scaling regime, which, in fact, emerges very quickly after a fast transition process. We show that the significant information hidden within the time series concerns memory induced by the solar cycle, as well as the power index $\mu$. The scaling parameter $\delta$ becomes a simple function of $\mu$, when memory is annihilated. Thus, the three walking rules yield a unique and precise value of $\mu$ if the memory is wisely taken under control, or cancelled by shuffling the data. All this makes compelling the conclusion that $\mu = 2.138 \pm 0.01$.Comment: 23 pages, 13 figure