The Poissonian Origin of Power Laws in Solar Flare Waiting Time Distributions

In this study we aim for a deeper understanding of the power law slope, $\alpha$, of waiting time distributions. Statistically independent events with linear behavior can be characterized by binomial, Gaussian, exponential, or Poissonian size distribution functions. In contrast, physical processes with nonlinear behavior exhibit spatio-temporal coherence (or memory) and "fat tails" in their size distributions that fit power law-like functions, as a consequence of the time variability of the mean event rate, as demonstrated by means of Bayesian block decomposition in the work of Wheatland et al.~(1998). In this study we conduct numerical simulations of waiting time distributions $N(\tau)$ in a large parameter space for various (polynomial, sinusoidal, Gaussian) event rate functions $\lambda(t)$, parameterized with an exponent $p$ that expresses the degree of the polynomial function $\lambda(t) \propto t^p$. We derive an analytical exact solution of the waiting time distribution function in terms of the incomplete gamma function, which is similar to a Pareto type-II function and has a power law slope of $\alpha = 2 + 1/p$, in the asymptotic limit of large waiting times. Numerically simulated random distributions reproduce this theoretical prediction accurately. Numerical simulations in the nonlinear regime ($p \ge 2$) predict power law slopes in the range of $2.0 \le \alpha \le 2.5$. The self-organized criticality model yields a prediction of $\alpha=2$. Observations of solar flares and coronal mass ejections (over at least a half solar cycle) are found in the range of $\alpha_{obs} \approx 2.1-2.4$. Deviations from strict power law functions are expected due to the variability of the flare event rate $\lambda(t)$, and deviations from theoretically predicted slope values $\alpha$ occur due to the Poissonian weighting bias of power law fits.