Exact Calculation of the Probabilities of Rare Events in Cluster-Cluster Aggregation

Area of Science:

• Statistical Physics
• Chemical Physics
• Computational Physics

Background:

• Cluster-cluster aggregation is a fundamental process in materials science and physics.
• Calculating probabilities of rare events in such systems is computationally challenging.
• Existing methods often lack the precision for arbitrary collision kernels.

Purpose of the Study:

• To develop a novel action formalism for calculating rare event probabilities in cluster-cluster aggregation.
• To establish a pathwise large deviation principle for quantifying these rare events.
• To apply the formalism to specific collision kernels (constant, sum, product) and analyze the rate function and optimal trajectories.

Main Methods:

• Development of an action formalism based on large deviation theory.
• Establishment of a pathwise large deviation principle where the total mass represents the rate.
• Exact calculation of the rate function and optimal evolution trajectory for constant, sum, and product kernels.

Main Results:

• The action formalism successfully calculates rare event probabilities for arbitrary collision kernels.
• Exact solutions for the rate function and optimal trajectories were obtained for constant, sum, and product kernels.
• A discontinuity in the second derivative of the rate function was identified for the product kernel.
• Theoretical predictions align with specialized simulations for rare event calculations.

Conclusions:

• The developed action formalism provides a powerful and exact tool for analyzing rare events in cluster-cluster aggregation.
• The findings offer precise insights into system dynamics, particularly for the product kernel's rate function.
• The agreement with simulations validates the theoretical approach for rare event probability calculations.