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arXiv · 1802.09440

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

Loïc Turban
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Abstract

We study the two-species diffusion-annihilation process, $A+B\rightarrow$ \O, 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.

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Loïc Turban. 2018-02-26. Two-species diffusion-annihilation process on the fully-connected lattice: probability distributions and extreme value statistics. https://doi.org/10.1088/1751-8121/aaccc1

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