  The process of numerical thermalization in particle-in-cell (PIC) simulations
has been studied extensively. It is analogous to Coulomb collisions in real
plasmas, causing particle velocity distributions (VDFs) to evolve towards a
Maxwellian as macroparticles experience polarization drag and resonantly
interact with the fluctuation spectrum. This paper presents a practical
tutorial on the effects of numerical thermalization in 2D PIC applications.
Scenarios of interest include simulations which must be run for many thousands
of plasma periods and contain a population of cold electrons that leave the
simulation space very slowly. This is particularly relevant to many low
temperature plasma discharges and materials processing applications. We present
numerical drag and diffusion coefficients and their associated timescales for a
variety of grid resolutions, discussing the circumstances under which the
electron VDF is modified by numerical thermalization. Though the effects
described here have been known for many decades, direct comparison of
analytically derived, velocity-dependent numerical relaxation timescales to
those of other relevant processes has not often been applied in practice due to
complications that arise in calculating thermalization rates in 1D simulations.
Using these comparisons, we estimate the impact of numerical thermalization in
several example low temperature plasma applications including capacitively
coupled plasma (CCP) discharges, inductively coupled plasma (ICP) discharges,
beam plasmas, and hollow cathode discharges. Finally, we discuss possible
strategies for mitigating numerical relaxation effects in 2D PIC simulations.
