Monte Carlo simulation of particle interactions at high dynamic range: Advancing beyond the Googol
We present a method which extends Monte Carlo studies to situations that require a large dynamic range in particle number. The underlying idea is that, in order to calculate the collisional evolution of a system, some particle interactions are more important than others and require more resolution, while the behavior of the less important, usually of smaller mass, particles can be considered collectively. In this approximation groups of identical particles, sharing the same mass and structural parameters, operate as one unit. The amount of grouping is determined by the zoom factor -- a free parameter that determines on which particles the computational effort is focused. Two methods for choosing the zoom factors are discussed: the `equal mass method,' in which the groups trace the mass density of the distribution, and the `distribution method,' which additionally follows fluctuations in the distribution. Both methods achieve excellent correspondence with analytic solutions to the Smoluchowski coagulation equation. The grouping method is furthermore applied to simulations involving runaway kernels, where the particle interaction rate is a strong function of particle mass, and to situations that include catastrophic fragmentation. For the runaway simulations previous predictions for the decrease of the runaway timescale with the initial number of particles ${\cal N}$ are reconfirmed, extending ${\cal N}$ to $10^{160}$. Astrophysical applications include modeling of dust coagulation, planetesimal accretion, and the dynamical evolution of stars in large globular clusters. The proposed method is a powerful tool to compute the evolution of any system where the particles interact through discrete events, with the particle properties characterized by structural parameters.