In this paper we analyze the noise in macro-particle methods used in plasma physics and fluid dynamics, leading to approaches for minimizing the total error, focusing on electrostatic models in one dimension. We begin by describing kernel density estimation for continuous values of the spatial variable x, expressing the kernel in a form in which its shape and width are represented separately. The covariance matrix $C(x,y)$ of the noise in the density is computed, first for uniform true density. The bandwidth of the covariance matrix is related to the width of the kernel. A feature that stands out is the presence of constant negative terms in the elements of the covariance matrix both on and off-diagonal. These negative correlations are related to the fact that the total number of particles is fixed at each time step; they also lead to the property $\int C(x,y)dy=0$. We investigate the effect of these negative correlations on the electric field computed by Gauss's law, finding that the noise in the electric field is related to a process called the Ornstein-Uhlenbeck bridge, leading to a covariance matrix of the electric field with variance significantly reduced relative to that of a Brownian process.

For non-constant density, $\rho (x)$, still with continuous x, we analyze the total error in the density estimation and discuss it in terms of bias-variance optimization (BVO). For some characteristic length l, determined by the density and its second derivative, and kernel width h, having too few particles within h leads to too much variance; for h that is large relative to l, there is too much smoothing of the density. The optimum between these two limits is found by BVO. For kernels of the same width, it is shown that this optimum (minimum) is weakly sensitive to the kernel shape.

We repeat the analysis for x discretized on a grid. In this case the charge deposition rule is determined by a particle shape. An important property to be respected in the discrete system is the exact preservation of total charge on the grid; this property is necessary to ensure that the electric field is equal at both ends, consistent with periodic boundary conditions. We find that if the particle shapes satisfy a partition of unity property, the particle charge deposited on the grid is conserved exactly. Further, if the particle shape is expressed as the convolution of a kernel with another kernel that satisfies the partition of unity, then the particle shape obeys the partition of unity. This property holds for kernels of arbitrary width, including widths that are not integer multiples of the grid spacing.

We show results relaxing the approximations used to do BVO optimization analytically, by doing numerical computations of the total error as a function of the kernel width, on a grid in x. The comparison between numerical and analytical results shows good agreement over a range of particle shapes.

We discuss the practical implications of our results, including the criteria for design and implementation of computationally efficient particle shapes that take advantage of the developed theory.

在本文中，我们分析了用于等离子体物理和流体动力学的大粒子方法中的噪声，从而提出了将总误差最小化的方法，重点是在一维静电模型上。我们从描述空间变量x的连续值的核密度估计开始，以核的形状和宽度分别表示的形式表示核。协方差矩阵$C（X，ÿ）$首先，对于均匀的真实密度，计算密度中的噪声的平方。协方差矩阵的带宽与内核的宽度有关。突出的特征是在对角线内和对角线外协方差矩阵的元素中都存在恒定的负项。这些负相关与以下事实有关：每个时间步长的粒子总数都是固定的。他们也导致财产$\int C（X，ÿ）dÿ=0$。我们研究了这些负相关对高斯定律计算出的电场的影响，发现电场中的噪声与一个称为Ornstein-Uhlenbeck桥的过程有关，从而导致电场的协方差矩阵，且方差显着减小相对于布朗过程。

对于非恒定密度， $\rho （X）$仍然使用连续的x，我们分析密度估计中的总误差，并根据偏差方差优化（BVO）进行讨论。对于某些特征长度l（由密度及其二阶导数确定）和内核宽度h，h内的粒子太少会导致太大的变化；反之，对于相对于l较大的h，密度的平滑度太大。这两个极限之间的最佳值由BVO确定。对于相同宽度的籽粒，表明该最佳值（最小值）对籽粒形状较不敏感。

我们对网格上离散的x重复分析。在这种情况下，电荷沉积规则由颗粒形状决定。在分立系统中要考虑的一个重要属性是精确保留电网上的总电荷。此特性对于确保两端的电场均等（与周期性边界条件一致）是必不可少的。我们发现，如果粒子形状满足单位性质的划分，则沉积在网格上的粒子电荷将被完全守恒。此外，如果将粒子形状表示为一个核与另一个满足单位分配的核的卷积，则粒子形状服从单位分配。此属性适用于任意宽度的内核，包括不是网格间距的整数倍的宽度。

通过在x的网格上进行总误差作为内核宽度函数的数值计算，我们展示了结果放松了用于进行BVO优化分析的近似值。数值结果与分析结果之间的比较表明，在一定范围的颗粒形状上具有良好的一致性。

我们讨论了结果的实际含义，包括设计和实现利用发达理论的高效计算粒子形状的标准。