Emerging extreme value and Fermi-Dirac distributions in the Lévy branching and annihilating process

We study the dynamics of the branching and annihilating process with long-range interactions. Static particles generate an offspring and annihilate upon contact. The branching distance is supposed to follow a Lévy-like power-law distribution with P(r)∝1/r^{α}. We analyze the long term behavior of the mean particles number and its fluctuations as a function of the parameter α that controls the range of the branching process. We show that the dynamic exponent associated with the particle number fluctuations varies continuously for α<4 while the particle number exponent only changes for α<3. A crossover from extreme value Frechet (at α=3) and Gumbell (for 2<α<3) distributions is developed, similar to the one reported in recent experiments with cw-pumped random fiber lasers presenting underlying gain and Lévy processes. We report the dependence of the relevant dynamical power-law exponents on α showing that explosive growth takes place for α≤2. Further, the average occupation number distribution is shown to evolve from the standard Fermi-Dirac form to the generalized one within the context of nonextensive statistics.