论文标题
在一维Jellium模式下线性统计方差的确切公式
An exact formula for the variance of linear statistics in the one-dimensional jellium mode
论文作者
论文摘要
我们将$ n $颗粒的Jellium模型在外部谐波电位限制的线上，并具有成对的一维库仑强度的强度$α> 0 $。使用库仑气体方法，我们研究$ s =（1/n）\ sum_ {i = 1}^n f（x_i）$的统计信息，其中$ f（x）$原则上是一个任意的平滑函数。虽然$ s $的平均值易于计算，但由于远程库仑相互作用，差异是不平凡的。在本文中，我们证明了这种平均值的波动是高斯，具有差异$ {\ rm var}（s）\大约$ n $的b/n^3 $。我们为常数$ b = 1/（4α）\ int _ { - 2α}^{2α} [f'（x）]^2 \，dx $提供了精确的紧凑公式。此外，我们还计算出表征完整分布的尾巴$ {\ cal p}（s，n）$的完整大偏差功能，以$ f（x）$的几个不同示例。我们的分析预测通过数值模拟证实。
We consider the jellium model of $N$ particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength $α> 0$. Using a Coulomb gas method, we study the statistics of $s = (1/N) \sum_{i=1}^N f(x_i)$ where $f(x)$, in principle, is an arbitrary smooth function. While the mean of $s$ is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance ${\rm Var}(s) \approx b/N^3$ for large $N$. We provide an exact compact formula for the constant $b = 1/(4α) \int_{-2 α}^{2α} [f'(x)]^2\, dx$. In addition, we also calculate the full large deviation function characterising the tails of the full distribution ${\cal P}(s,N)$ for several different examples of $f(x)$. Our analytical predictions are confirmed by numerical simulations.
