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 $P\left(r\right)\propto 1/r^{\alpha }$. We analyze the long term behavior of the mean particles number and its fluctuations as a function of the parameter $\alpha$ that controls the range of the branching process. We show that the dynamic exponent associated with the particle number fluctuations varies continuously for $\alpha <4$ while the particle number exponent only changes for $\alpha <3$. A crossover from extreme value Frechet (at $\alpha =3$) and Gumbell (for $2<\alpha <3$) 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 $\alpha$ showing that explosive growth takes place for $\alpha \le 2$. 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.

• Received 13 January 2020
• Revised 20 March 2020
• Accepted 6 May 2020

DOI:https://doi.org/10.1103/PhysRevE.101.052136