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

Abstract

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.

Characterizing time intermittency in non-diffusive fast ion transport through plasma turbulence

“…By then combining equations (6) and (7), we arrive at an expression for the predicted on-phase skewness [19] mm, which corresponds well to results from simulations [21]. However, the fitted parameter ρ then underestimates the true width of the meanprofiles by a factor of ≈ 2.…”

“…The physical origin and extent of the displacements of this instantaneous fast ion beam have been quantified in detail by CS analysis. We have therefore developed an analytical model [21] for the statistics of time-series, where the time-average current density profile J(R, Z) = J(R) is constructed through the motion of a smaller (and approximately rigid) instantaneous profile j(R).…”

“…As illustrated in figure 9, the statistics of this motion are contained in a displacement PDF f (R), so that the timeaverage profile at a particular measurement position R 0 is given by the convolution of f and j. While the time-average J(R 0 ) corresponds to the 1st moment of a time-series taken at R 0 , this directly generalizes to higher orders, such that the qth order non-central moments J q (R 0 ) are given by Quantities such as skewness (or kurtosis) of local timeseries have been modeled quite successfully by adapting this method to simulations [21] as well as the experimental timeseries data [19], as described next.…”

“…No further assumptions on the shape of f (R) are required. From this we can again derive simple expressions to predict the central moments J c,q (see [19,21])…”

“…Conditional sampling (CS) is used on time-resolved Langmuir-probe (LP) measurements [20] to directly illustrate the effects of turbulent E × B-drifts as fast ion transport mechanism [17]. In section 4, we briefly review an analytical model developed in [21], that predicts the skewness of local fast ion time-series by modeling how the time-average fast ion profile is constructed through the motion of a smaller instantaneous fast ion beam, as described in detail in [19]. In section 5 we use the established picture of the meandering instantaneous fast ion beam, to motivate how intermittency can be quantified by the skewness of a simple, two-valued time-series.…”