Reaction Diffusion systems with Annihilation Mechanisms
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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.
Beskrivning
Ämne/nyckelord
Reaction-Diffusion Systems, Monte Carlo Simulation, Master Equation