Mykhailo Potomkin

Leonid Berlyand, Robert Creese, Pierre-Emmanuel Jabin et al.

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.

Leonid Berlyand, Pierre-Emmanuel Jabin, Mykhailo Potomkin

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.