Complexity reduction in many particles systems with random initial data

TL;DR

This paper introduces a novel computational method based on BBGKY hierarchy truncation to efficiently simulate large interacting particle systems with randomness, capturing correlations beyond mean field approximations.

Contribution

It develops a new approach that simultaneously reduces computational complexity, accounts for correlations, and handles randomness in many-particle systems.

Findings

Successfully captures emergent correlations in particle patterns

Reduces computational cost compared to direct simulations

Provides numerical solutions for nonlinear, non-local systems

Abstract

We consider the motion of interacting particles governed by a coupled system of ODEs with random initial conditions. Direct computations for such systems are prohibitively expensive due to a very large number of particles and randomness requiring many realizations in their locations in the presence of strong interactions. While there are several approaches that address the above difficulties, none addresses all three simultaneously. Our goal is to develop such a computational approach in order to capture the experimentally observed emergence of correlations in the collective state (patterns due to strong interactions). Our approach is based on the truncation of the BBGKY hierarchy that allows one to go beyond the classical Mean Field limit and capture correlations while drastically reducing the computational complexity. Finally, we provide an example showing a numerical solution of this nonlinear and non-local system.