Quasi-periodic boundary conditions for hierarchical algorithms used for the calculation of inter-particle electrostatic interactions

In the present work, a simplified way to apply tri-periodic Boundary Conditions (BCs) in hierarchical algorithms has been devised. The application of these BCs is demonstrated using an in-house algorithm that is used for the calculation of electrostatic interactions in a system of charged particles. The proposed quasi-periodic BCs entail a truncation of the infinite periodic domain with a reasonable cut-off error. For a correct representation of the physics, two properties have to be ensured: the convergence of the electrostatic forces and the isotropy of the electric field. The developed algorithm allows for a rather efficient and precise calculation of the former in a tri-periodic computational domain by separating them in short- and long-range parts, which are calculated exactly and approximately, respectively. The approximation error, computational cost and performance of the proposed algorithm are documented and thoroughly analyzed. Then, an application of the method is presented for dry mono-charged (all particles bear equal charge) particle flows where the fundamentals physics of particle-particle electrostatic interactions are investigated via characteristic length and time scales. It is shown that the underlying mechanism of these interactions is the Coulomb collision, a concept that allows to interpret these interactions in a rather intuitive way. In addition, an attempt is made to provide analytical estimations for several statistical quantities via dimensional analysis based on measured simulation data. Finally, the particle-induced electric field is presented and its characteristics are related to particle motion.