A new framework to study rare events in aggregation
October 7, 2024 | Bharti Dharapuram
A new theoretical framework to study rare events in aggregation uses a model that specifies how clusters form and evolve when particles collide with each other.
Researchers have developed an analytical framework for studying rare events in aggregation, a phenomenon that drives the formation of clouds and coagulation of proteins. This work was carried out by R Rajesh and V Subashri from The Institute of Mathematical Sciences, Chennai and Oleg Zaboronski from the University of Warwick. Their findings can be used to study diverse phenomena in environment and climate, biology, and materials science.
Aggregation occurs when particles collide and coalesce to form clusters. Existing theory describes the mean number of clusters at a given time but does not predict the probability of their occurrence and evolution over time. The authors used a branch of probability theory dealing with rare events, known as large deviation theory, as the framework of their study. They modelled a collection of particles and how their mass distribution evolves over time due to aggregation events. They considered three scenarios, which differ in how the rate of collisions depends on the colliding masses. For each of these, they derived the probability of occurrence of rare events and their most probable path, verifying their predictions using simulations.
“Aggregation has been studied using a mean field rate equation called the Smoluchowski equation for a long time,” Subashri says. In one of the scenarios where the equation has been applied, particles coalesce to form a gel that drastically changes the nature of their interactions. “The Smoluchowski equation predicts gelation, but does not capture these post-gelation dynamics,” she adds, emphasizing an important gap addressed by their study.
“We start with a master equation for a probability distribution, which describes how particles enter a configuration and how they exit it,” Subashri says about the analytical process. When the system is modelled several thousands of times it follows a typical path, but the group was interested in rare events that deviate from the usual. “It is hard to find the probability of rare events since there aren’t many established methodologies to do it,” she adds. The group used path integral methods to model these probabilities. “We had to do some modifications when gelation comes into the picture,” she adds. Results from decades ago gave them the clues to make the necessary adjustments.
“This is a step forward because it provides a new method by which one can analyze aggregation,” Subashri says. Their findings may allow us to study rare aggregation events, such as extreme climatic events and neurodegenerative diseases, with important human consequences. In such applications, we are modeling a physical system, which may not be that simple, Subashri cautions. “The current analytical framework provides us a guideline to proceed further.”
Reference: Rajesh, R., Subashri, V., & Zaboronski, O. (2024). Exact calculation of the probabilities of rare events in cluster-cluster aggregation. Physical Review Letters, 133(9), 097101. https://doi.org/10.1103/PhysRevLett.133.097101
