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, nonvanishing 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.

Figure 1

One-dimensional case. A rigid particle bunch, of density $n\left(x\right)$, moves along $x$ such that at time $t_k$ its center is at $x={\chi }_k$ (dashed lines). A detector located at $x=X$ counts the number of particles within a small region of effective length $L_{\text{eff}}$ (gray shade), leading to a TSN with $K$ samples. The mean of the TSN, $〈N\left(X\right)〉$, converges to the AC at $X$ for sufficiently large $K$.

Figure 2

Comparison of simulated values of TSN statistics (circles) to predictions (lines) of the mean (a), standard deviation (b), and skewness (c) at different detector locations $X$. We use ${\text{c}}_m=0$, ${\text{w}}_m=1$ [apparent from the width of the Gaussian profile in (a)], ${\text{w}}_n=0.7$, and ${\text{w}}_L=0.2$. Notice the large range of values of the skewness, which includes negative numbers close to $X=0$.

Figure 3

Using ${\text{w}}_m=1$, we compute the variance and the skewness of numerically generated TSNs $\left\{N_k\left(X\right)\right\}$ for different values of ${\text{w}}_{\mathit{nL}}$. (a) Simulated standard deviation plotted against the simulated mean (circles). Both the standard deviation and the mean have been normalized by $L_{\text{eff}}N$. This makes it simple to compare the results against predictions (lines) and highlights the fact that TSN moments can be determined at a particular detector location $X$ from knowledge of $〈N\left(X\right)〉$ and $L_{\text{eff}}N$. (b) Skewness versus normalized mean. No normalization is performed for the skewness as its value is not affected by rescalings of the $N_k\left(X\right)$.

Figure 4

Numerically generated covariance (circles) of the time series at two locations $X_1$ and $X_2$. The different colors correspond to different values of $X_1$ as shown in the legend. We use here ${\text{c}}_m=0$, ${\text{w}}_m=1$, and ${\text{w}}_{\mathit{nL}}=0.5$. The predicted covariances (lines) precisely follow the numerical results.

Figure 5

Normalized standard deviation (a) and skewness (b) of the noisy TSN $\left\{{\mathcal{N}}_k\left(X\right)\right\}$, plotted against the normalized mean, for varying levels of noise (as indicated in the legend). We assume Gaussian-shaped $n\left(x\right)$, $L\left(x\right)$, and $〈N\left(x\right)〉$, as in Fig. 2, with parameters ${\text{c}}_m=0$, ${\text{w}}_m=1$, ${\text{w}}_n=0.7$, and ${\text{w}}_L=0.2$, and Gaussian noise with ${\mu }_{\eta }$, ${\sigma }_{\eta }$ independent of the detector location $X$ (clearly ${\gamma }_{\eta }=0$). These predictions are obtained from Eqs. (19), (20), and (21). Notice that even small values of noise force the skewness to fall to zero in regions of weak signal, i.e., where $〈\mathcal{N}\left(X\right)〉-{\mu }_{\eta }\left(X\right)=0$.

Figure 6

(a) Relative error between the predicted variance [RHS in Eq. (9)] using approximations [i.e., using Eq. (25)] and using exact expressions of $m_q$ [Eq. (14)], for the case with no noise and Gaussian $m\left(X\right)$. The error is computed as ${\text{E}}_{\text{rel}}\left(\text{Var}\right)=[{\text{Var}}_{\text{appr}}\left(X\right)-{\text{Var}}_{\text{exact}}\left(X\right)]/{\text{Var}}_{\text{exact}}\left(X\right)$ and analogously for the skewness. (b) Relative error between predicted skewness using approximations and using exact expressions. Notice that these results are valid for any ${\text{w}}_m>0$. Indeed, the expressions for the errors can be cast in a form that only depends on ${\text{w}}_{\mathit{nL}}/{\text{w}}_m$ and $\sqrt{2\pi }{\text{w}}_{m}m\left(X\right)$.

Figure 7

Error in the covariance when using the approximation [Eq. (26)] compared to the exact result [Eq. (17)] for Gaussian-shaped $m$ (${\text{c}}_m=0$, ${\text{w}}_m=1$) and $n_L$ (${\text{w}}_{\mathit{nL}}=0.2$). Each marker represents one choice of $X_1$, $X_2$, where $X_1=0,0.4,1.2,2$, or 3 and $X_2\in [-5,5]$. To better compare values near zero crossings, we use the symmetric absolute error ${\text{E}}_{\text{sae}}\left(\text{Cov}\right)=2\left|{\text{Cov}}_{\text{appr}}-{\text{Cov}}_{\text{exact}}\right|/\left(\left|{\text{Cov}}_{\text{appr}}\right|+\left|{\text{Cov}}_{\text{exact}}\right|\right)$ instead of the more common relative error. An ${\text{E}}_{\text{sae}}\le 0.1=10\%$ is observed for all values of the approximate covariance ${\text{Cov}}_{\text{appr}}$ except very close to zero, where small denominators make the error estimation impractical.

Figure 8

(a) Some possible shapes for $n_L$ that fulfill the requirements for the approximations in Sec. 3a. All shapes are correctly normalized to integrate to 1 and have ${\text{w}}_{\mathit{nL}}=0.15$. (b) Application of Eq. (31) to numerically generated TSNs using the $n_L\left(x\right)$ shown in (a), $f\left(x\right)=\text{G}(x;0,1-{\text{w}}_{\mathit{nL}}^2)$, and $K={10}^5$ gives the results shown with circles [same color labeling as in (a)]. The detector location is $X=0$ in all cases. Gaussian noise is added with ${\mu }_{\eta }=0$ and ${\sigma }_{\eta }=0.2\times N^{\text{max}}$, and $N^{\text{max}}$ is determined from the known parameters (it is not estimated from the TSNs). The agreement with the expressions for $b\left(q\right)$ in Eq. (28) (dashed lines) allows one to distinguish shapes based on information gathered from a single TSN.

Figure 9

Two-dimensional case. A rigid particle bunch (black) of density $n\left(\mathbf{r}\right)$ moves on the $xy$ plane such that its center at time $t_k$ is located at $\mathbf{r}={\mathbf{\nu }}_k=({\chi }_k,{\psi }_k)$. A detector at $\mathbf{r}=\mathbf{R}=(X,Y)$ counts the number of particles within a region of effective area $A_{\text{eff}}$ (shaded light gray) near $\mathbf{R}$. At large $K$, the detector will have recorded an AC $〈N\left(\mathbf{R}\right)〉$.

Figure 10

Fast ion propagation in TORPEX. The toroidal vessel contains a hydrogen plasma in an SMT configuration with open helical magnetic field lines. A Li-6+ source injects fast lithium ions into the plasma. These ions then propagate towards a detector positioned at a different toroidal location. Interaction with the turbulent plasma potential (depicted at two toroidal locations, one near the source and the other near the detector) affects the ion trajectories and introduces variations in the number of ions that reach the detector. The detector can be moved in the $x$ and $y$ directions (indicated in white) to perform measurements at different locations on the same $xy$ (poloidal) plane.

Figure 11

Some fast ion bunches as they arrive at the $xy$ (poloidal) plane of the detector. Different bunches (shaded structures) arrive at different locations after propagating through the turbulent plasma. Darker parts correspond to higher density of ions, as per the color bar. Comparison with the shape and location of the bunches as injected at the source (indicated in the figure) shows that fast ions experience drifts as well as shear. Once the effects of the detector are added, $〈N\left(\mathbf{r}\right)〉$ can be computed. Contour lines of the result (blue dashed lines) show a time-averaged profile that is much wider than the individual bunches.

Figure 12

Fast-ion simulation results. TSNs are determined for $\approx 100$ choices of detector location (“+” markers) and the mean, variance, and skewness are computed. The results are then linearly interpolated between the markers and the outcome is plotted following the color map in each column. TSNs are dimensionless, as they hold data in units of number of particles. (a) Mean, (b) variance, and (c) skewness for the case of no noise. (d), (e), (f) Same statistics when Gaussian noise with zero mean and ${\sigma }_{\eta }={10}^3$ is added to all TSNs. Even though the mean and variance remain similar, a comparison of (c) and (f) confirms that low levels of noise can have a big impact on the skewness.

Figure 13

Normalized variance (a) and skewness (b) of the simulated fast ion TSNs, plotted against the mean. Gaussian noise with zero mean and ${\sigma }_{\eta }={10}^3$ has been added to all TSNs. Each black dot corresponds to statistics computed for a single TSN from one particular detector location (“+” markers in Fig. 12). The error bars are $1\sigma$ uncertainties estimated using simple bootstrapping with nonoverlapping blocks of 500 samples from the corresponding TSN (changing the block size does not appreciably alter the outcome). The red dashed curve in (a) is a fit of the data using Eq. (43) as a model. The fit allows us to determine the parameter $n_A^{\text{max}}$, which can then be replaced in Eq. (44) to produce a predicted skewness curve [red dotted line in (b)] for an assumed Gaussian-shaped bunch. There is good agreement between the model and the simulations.