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Meandering particle bunches and a link between averages of time series of particle counts and higher-order moments

Time-averaged spatially resolved measurements are used in many fields of physics to determine spatial distributions of a physical quantity. Although one could think that time-averaging suppresses all information on time variation, there are some situations in which a link can be established between time-averaging and time-variability. In this paper, we consider a simple system composed of a particle bunch that moves in space without deforming, and a detector placed at a point in space. The detector continuously counts the number of particles in its neighborhood. Upon sampling, the detector signal gives rise to a time-series with, in general, non-vanishing variance. Time-series obtained by placing the detector at different locations can then be used to obtain a time-average distribution of the number of particles by computing the time-average of all the time-series. We show that there is a close relationship between this average profile and higher order statistics of the time-series, including the variance and skewness. We also show a simple procedure by which individual time-series can be used to determine features of the shape of the particle bunch.