Two-species diffusion-annihilation process on the fully-connected lattice: probability distributions and extreme value statistics

Abstract

We study the two-species diffusion-annihilation process, $A+B\to$ Ø, on the fully-connected lattice. Probability distributions for the number of particles and the reaction time are obtained for a finite-size system using a master equation approach. Mean values and variances are deduced from generating functions. When the reaction is far from complete, i.e., for a large number of particles of each species, mean-field theory is exact and the fluctuations are Gaussian. In the scaling limit the reaction time displays extreme-value statistics in the vicinity of the absorbing states. A generalized Gumbel distribution is obtained for unequal initial densities, ${\rho }_A>{\rho }_B$. For equal or almost equal initial densities, ${\rho }_A\simeq {\rho }_B$, the fluctuations of the reaction time near the absorbing state are governed by a probability density involving derivatives of ${\vartheta }_4$, the Jacobi theta function.

30 pages, 18 figures. Continuation of arXiv:1711.01248. published version

References in corpus (2)

• Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems
• Records for the number of distinct sites visited by a random walk on the fully-connected lattice