Klimontovich, Boltzmann, and Vlasov – What are PIC Codes Really Solving?

The derivation of the particle-in-cell (PIC) method is typically performed by noting that the particle equations of motion remain the same if one maintains the same charge-to-mass (Q/M) ratio of the particle – this allows the introduction of macroparticles to greatly save the amount of computational work needed to simulate a given particle distribution, albeit at the cost of increased particle noise. It is also well-known that the derivation of the Vlasov equation from the plasma kinetic equation can be done by hypothetically breaking the charged particles into many smaller pieces, again while maintaining the charge-to-mass ratio, which will remove the discrete-particle nature of the plasma and allow the definition of smooth distribution function [1]. As the number of pieces tends to infinity, we find that this pulverization limit yields the Vlasov equation, or the “collisionless” Boltzmann equation. As PIC is typically considered to be approximating the Vlasov equation, it seems that these two constructs, namely pulverization and macroparticles, might be in some kind of conflict.