Kinetics of fragmentation-annihilation processes

TL;DR

This paper analyzes the kinetics of fragmentation-annihilation systems, revealing how mass loss and catalyst interactions influence long-term behavior and decay patterns in particle populations.

Contribution

It introduces a continuous mean-field model for fragmentation-annihilation processes with nonlinear and catalyst-controlled reactions, exploring complex asymptotic regimes.

Findings

Small-mass cluster annihilation dominates asymptotic behavior.

Decay patterns can be non-universal and depend on catalyst densities.

Scaling regimes are influenced by average mass and mass-density dynamics.

Abstract

We investigate the kinetics of systems in which particles of one species undergo binary fragmentation and pair annihilation. In the latter, nonlinear process, fragments react at collision to produce an inert species, causing loss of mass. We analyse these systems in the reaction-limited regime by solving a continuous model within the mean-field approximation. The rate of fragmentation, for a particle of mass $x$ to break into fragments of masses $y$ and $x-y$, has the form $x^{\lambda-1}$ ($\lambda>0$), and the annihilation rate is constant and independent of the masses of the reactants. We find that the asymptotic regime is characterized by the annihilation of small-mass clusters. The results are compared with those for a model with linear mass-loss (i.e.\ with a sink). We also study more complex models, in which the processes of fragmentation and annihilation are controlled by mutually-reacting catalysts. Both pair- and linear-annihilation are considered. Depending on the specific model and initial densities of the catalysts, the time-decay of the cluster-density can now be very unconventional and even non-universal. The interplay between the intervening processes and the existence of a scaling regime are determined by the asymptotic behaviour of the average-mass and of the mass-density, which may either decay indefinitely or tend to a constant value. We discuss further developments of this class of models and their potential applications.