Double power-law universal scaling function for the distribution of waiting times in labquake catalogs

We postulate that waiting times between avalanches in self-organized critical systems are distributed according to a universal double power-law probability density. This density is defined by two critical exponents α and β characterizing the distribution of short (∼δ^{-α}) and long (∼δ^{-β}) waiting times, and a crossover parameter δ_{0} that separates the two behaviors in a sharp shoulder. This crossover parameter depends on the system properties as well as on the observation conditions. It can be used as a scaling factor that transforms the distributions into a universal scaling law as proposed by Per Bak. We use experimental data from labquake catalogs (acoustic emission events) obtained during the uniaxial compression of a number of charcoal samples with different hardnesses and different energy thresholds. To obtain good fits it is essential that the catalogs are long enough to include a representative critical mixture of periods with different avalanche rates. In all the cases studied, individual maximum likelihood analysis allows the exponents α and β and the crossover parameter δ_{0} to be fitted. This parameter shows a clear dependence with the energy threshold that can be explained from the Gutenberg-Richter law for the avalanche energy distributions. The observed variations of the exponents α and β fall within the sample-to-sample variability, which suggest that these values could be universal. We estimate mean values α=0.9±0.1 and β=2.0±0.3 from the full set of recorded experimental data. These values are close to the combination α=1, β=2, which exhibits a special mathematical cancellation of singularities.