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Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique

We derive the kinetic theory of fluctuations in physically and numerically stable particle-in-cell (PIC) simulations of electrostatic plasmas. The starting point is the single-time correlations at the simulation start between the statistical fluctuations of weighted densities of macroparticle centers in the plasma particle phase-space. The single-time correlations at all time steps and in each spatial grid cell are then determined from the Laplace-Fourier transforms of the discretized Klimontovich-like equation for the macroparticles and Maxwell's equations for the fields as computed by modern PIC codes. We recover the expressions for the electrostatic field and the plasma particle density fluctuation autocorrelations spectra as well as the kinetic equations describing the average evolution of PIC-simulated plasma particles, first derived by Langdon in 1970, using a macroparticle test approach perturbing a discretized Vlasovian plasma and then averaging the obtained physical quantity over the initial macroparticle velocity distribution. We generalize and extend these results to the modern algorithms in PIC codes and using arbitrary macroparticle weights. Analytical estimates of statistical fluctuations single-time correlation amplitudes are derived as a function of the plasma simulation parameters, using the central limit theorem in the limit of a large number of macroparticles per cell. The theory is then used to analyze the ensemble averaging technique of PIC simulations where statistical averages are performed over ensembles of PIC simulations, modeling the same plasma physics problem but using different statistical realizations of the initial distribution functions of the macroparticles.