Rare events in cluster-cluster aggregation [HBNI Th255] 

Abstract

To summarize, we developed a biased Monte Carlo algorithm to compute probabili- ties of rare events in irreversible cluster-cluster aggregation for an arbitrary collision kernel. In particular, the algorithm measures P (M, N, t), the probability of N par- ticles remaining at time t when there are M particles initially, as well as the most probable trajectories for fixed M , N , and t. By choosing appropriate biases, the algorithm can efficiently sample the tails of the distribution with low computational e↵ort. We prove that the algorithm is ergodic by specifying a protocol that trans- forms any given trajectory to a standard trajectory using valid Monte Carlo moves. The algorithm is benchmarked against the exact solution for the constant kernel. To characterize the algorithm, we define autocorrelation times ⌧t and ⌧Q , corre- sponding to the waiting times as well as the configurations. We find that ⌧t is much smaller than ⌧Q for almost the entire range of parameters. From simulations for di↵erent M , we find that ⌧t is at most only weakly dependent on M , while ⌧Q is pro- portional to M 2 . Based on the dependence of ⌧t and ⌧Q on the bias w, the fraction of particles remaining = N/M , and the parameter p which decides what fraction of the Monte Carlo moves are changes to configurations, we conclude that it is best to choose a value of p as close to 1 as possible. Generalizing the numerical results for constant, sum, and product kernels, we con- clude that there exists a large deviation principle for arbitrary kernels, where the total mass M is the rate. This provides hints for a more rigorous treatment of the large deviation function for the problem of aggregation. In the next chapter, we provide a derivation of the large deviation function for some kernels. Although this chapter deals with binary aggregation, the algorithm that we have developed can also be easily generalized to the numerical study of the non-binary processes kA ! `A, with suitably modified rates. Adding spatial degrees of freedom, and transport, like di↵usion, is a problem of interest. However, generalizing the algorithm to such systems is a challenging problem. Adding a competing process such as fragmentation is another problem of interest [12, 13, 58, 59].Competing processes like these can lead to phase transitions and oscillations, at least in the mean field limit [60]. These are promising areas for future study.