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We derive the kinetic theory of fluctuations in physically and numericallystable particle-in-cell (PIC) simulations of electrostatic plasmas. Thestarting point is the single-time correlations at the simulation start betweenthe statistical fluctuations of weighted densities of macroparticle centers inthe plasma particle phase-space. The single-time correlations at all time stepsand in each spatial grid cell are then determined from the Laplace-Fouriertransforms of the discretized Klimontovich-like equation for the macroparticlesand Maxwell's equations for the fields as computed by modern PIC codes. Werecover the expressions for the electrostatic field and the plasma particledensity fluctuation autocorrelations spectra as well as the kinetic equationsdescribing the average evolution of PIC-simulated plasma particles, firstderived by Langdon in 1970, using a macroparticle test approach perturbing adiscretized Vlasovian plasma and then averaging the obtained physical quantityover the initial macroparticle velocity distribution. We generalize and extendthese results to the modern algorithms in PIC codes and using arbitrarymacroparticle weights. Analytical estimates of statistical fluctuationssingle-time correlation amplitudes are derived as a function of the plasmasimulation parameters, using the central limit theorem in the limit of a largenumber of macroparticles per cell. The theory is then used to analyze theensemble averaging technique of PIC simulations where statistical averages areperformed over ensembles of PIC simulations, modeling the same plasma physicsproblem but using different statistical realizations of the initialdistribution functions of the macroparticles.

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