Continuum approximations to systems of correlated interacting particles

TL;DR

This paper compares different continuum approximation methods for systems of correlated interacting particles, demonstrating that the Truncation Approximation outperforms traditional methods in accuracy and computational efficiency.

Contribution

It introduces and evaluates the Truncation Approximation as a more accurate and stable alternative to existing continuum methods for particle systems.

Findings

KSA and TA outperform MFA in accuracy.

TA is more numerically stable than KSA.

TA is less computationally expensive than KSA.

Abstract

We consider a system of interacting particles with random initial conditions. Continuum approximations of the system, based on truncations of the BBGKY hierarchy, are described and simulated for various initial distributions and types of interaction. Specifically, we compare the Mean Field Approximation (MFA), the Kirkwood Superposition Approximation (KSA), and a recently developed truncation of the BBGKY hierarchy (the Truncation Approximation - TA). We show that KSA and TA perform more accurately than MFA in capturing approximate distributions (histograms) obtained from Monte Carlo simulations. Furthermore, TA is more numerically stable and less computationally expensive than KSA.