Numerical Thermalization Timescales in 1D, 2D, and 3D Electrostatic Particle-in-Cell Simulations: Considerations for Modeling Plasma Discharges Relevant to Materials Processing Applications

Numerical thermalization in particle-in-cell (PIC) simulations is often confused with numerical heating or cooling. However, it can result in the relaxation of particle velocity distribution functions (VDFs) on timescales much shorter than that of energy non-conservation. A numerical collision operator accounting for these effects is well-known. [1,2] However, kinetic blocking results in the main non-aliasing term in the numerical collision operator annihilating any stable VDF in 1D-PIC simulations. Thus, 1D-PIC numerical thermalization timescales are often very large and analytical expressions for these timescales are not always available.

As 2D- or even 3D-PIC simulations of plasma discharges become increasingly feasible and desired, numerical thermalization becomes increasingly relevant. Not only are the timescales generally smaller than in 1D-PIC, but the numerical collision operator provides analytical estimates of velocity-dependent timescales for VDF relaxation. In this work we examine these estimates and their applicability to PIC simulations relevant for materials processing applications.

[1] C. Birdsall and A. Langdon, Plasma Physics via Computer Simulation, Taylor and Francis, (2004)

[2] M. Touati, R. Codur, F. Tsung, V. K. Decyk, W. B. Mori and L. O. Silva, "Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique," Plasma Phys. Control. Fusion, 64, 115014 (2022)