Numerical noise due to binning in macroparticle simulations

Numerical noise is an important parameter to minimize to ensure convergence of the macroparticle simulation results. In the context of simulations done for longitudinal beam dynamics studies, the longitudinal line density is usually obtained by binning a macroparticle distribution to compute wake potentials. The noise due to the binning can lead to artificial emittance blow-up or erronously trigger beam instability too early. Convergence studies are presently done empirically, to verify that results are not drastically changing with the number of macroparticles. In this note, an analytical expression for the noise amplitude as a function of the bunch spectrum frequency for a stationary distribution is benchmarked against numerical calculations. The analytical expression is then used to express the relevant scaling laws of the numerical noise and define a basic criterion on the number of macroparticles to use for different cases.