Reaction Diffusion systems with Annihilation Mechanisms

Abstract

This thesis investigates the complex dynamics of single-species annihilation-diffusion systems using a Monte Carlo simulation model. The systems are governed by the reaction kA → ∅, where particles annihilate in k-tuples on a discrete lattice system. Special focus is directed towards the k = 2, 4 particle annihilation scenarios, aiming to comprehend the influence of factors such as system size, particles’ diffusivity, and the annihilation rate on the particle correlation. The developed simulation model has been validated against theoretical predictions from mean field theory (MFT) and existing literature, demonstrating its robustness in capturing the behavior of such systems. The research reveals that the system dynamics are impacted by the interplay between diffusion and annihilation rates. In particular, it is shown that the diffusion-dominant regime is reached for 2-particle annihilation when diffusion is more likely than annihilation. In contrast, for 4- particle annihilation, while the system continues to decrease particles at the MFT rate, the diffusion process induces a delay in the time to extinction. Findings suggest that the MFT, while generally effective, is inadequate for fully capturing the dynamics of these systems, especially in scenarios where diffusion becomes prevalent. The challenge lies in understanding why diffusion, which slows down the system, does not significantly affect the particle annihilation rate, and how this seemingly paradoxical behavior can be incorporated into the theoretical framework. This raises the question of refining the MFT to include diffusion effects or exploring the applicability of the Kolmogorov-Petrovsky-Piskunov equation in such cases. This research underscores the complex dynamics inherent in annihilation-diffusion systems and contributes to our understanding of how these systems behave under different conditions. The findings set the stage for further research, which may provide insights into more comprehensive and accurate modeling of these systems, especially when particle correlations are strong.