A Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation

TL;DR

This paper introduces a biased Monte Carlo algorithm capable of efficiently sampling extremely rare events in cluster-cluster aggregation, enabling accurate estimation of very low probabilities and large deviation functions across various kernels.

Contribution

The authors develop a novel ergodic Monte Carlo method that modifies collision sequences and waiting times to sample rare aggregation events with unprecedented low probabilities.

Findings

Successfully samples probabilities as low as 10^{-40}

Reproduces exact large deviation functions for constant kernels

Extends to gelling kernels and computes instanton trajectories

Abstract

We develop a biased Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels. Given a trajectory with a fixed number of collisions, the algorithm modifies both the waiting times between collisions, as well as the sequence of collisions, using local moves. We show that the algorithm is ergodic by giving a protocol that transforms an arbitrary trajectory to a standard trajectory using valid Monte Carlo moves. The algorithm can sample rare events with probabilities of the order of $10^{-40}$ and lower. The algorithm's effectiveness in sampling low-probability events is established by showing that the numerical results for the large deviation function of constant-kernel aggregation reproduce the exact results. It is shown that the algorithm can obtain the large deviation functions for other kernels, including gelling ones, as well as the instanton trajectories for atypical times. The dependence of the autocorrelation times, both temporal and configurational, on the different parameters of the algorithm is also characterized.