Meshfree particle model for kinetic plasma simulations

We revisit a meshfree particle model for kinetics of a 1D electrostatic

plasma, using kernel density estimation and a similar method for the electric

field E. The kernel K(x − y) represents the macroparticle charge distribution.

Two length scales enter, the width w of K and the interparticle spacing λ. This

model conserves momentum and energy. Similarly, continuity is satisfied exactly,

and the Gauss’s law and Ampere’s law formulations are exactly equivalent. A

unified analysis is used for numerical stability and noise properties. The force

can be computed directly using the convolution K₂ = K ∗ K, and K₂ is positive

definite. We discuss the analogy in the presence of a grid. We can specify a

single kernel K_p , related to the `kernel trick’ of machine learning. Numerical

instability can occur unless K_p is positive definite, related to a breakdown in

energy conservation. For the noise analysis, the covariance matrix for the electric field shows a plasma dispersion function modified by w and λ. The number of particles per cell does not enter, and the noise is characterized by the number of particles per kernel width, i.e. w/λ. We present the bias-variance optimization (BVO) for the electric field, and compare it to the density BVO.