Efficient simulation of stochastic interactions among representative Monte Carlo particles^★

¹ Institute of Theoretical Astrophysics, Center for Astronomy (ZAH), Ruprecht-Karls-Universität Heidelberg, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
e-mail: moritz.beutel@ziti.uni-heidelberg.de
² Institute of Computer Engineering (ZITI), Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

Received: 27 June 2023
Accepted: 20 December 2023

Abstract

Context. Interaction processes between discrete particles are often modelled with stochastic methods such as the Representative Particle Monte Carlo (RPMC) method which simulate mutual interactions (e.g. chemical reactions, collisions, gravitational stirring) only for a representative subset of n particles instead of all N particles in the system. However, in the traditionally employed computational scheme the memory requirements and the simulation runtime scale quadratically with the number of representative particles.

Aims. We want to develop a computational scheme that has significantly lower memory requirements and computational costs than the traditional scheme, so that highly resolved simulations with stochastic processes such as the RPMC method become feasible.

Results. In this paper we propose the bucketing scheme, a hybrid sampling scheme that groups similar particles together and combines rejection sampling with a coarsened variant of the traditional discrete inverse transform sampling. For a v-partite bucket grouping, the storage requirements scale with n and v², and the computational cost per fixed time increment scales with n ⋅ v, both thus being much less sensitive to the number of representative particles n. Extensive performance testing demonstrates the higher efficiency and the favourable scaling characteristics of the bucketing scheme compared to the traditional approach, while being statistically equivalent and not introducing any new requirements or approximations. With this improvement, the RPMC method can be efficiently applied not only with very high resolution but also in scenarios where the number of representative particles increases over time, and the simulation of high-frequency interactions (such as gravitational stirring) as a Monte Carlo process becomes viable.