Question

英语翻译，急 ！！！求高手进！ 不要在线翻译器！谢谢

PERIODIC BOUNDARY CONDITIONS
We have also developed a version of the code with periodic boundary conditions for evaluating the potential energy of a set of charges in a cube and its infinite periodic images. The usual procedure, involving Ewald sums, is well documented in many texts. The Ewald sums over all infinite periodic images of each charge can be carried out in a time proportional to N 3/2.
In the fast multipole method, the procedure is initiated in the same manner as in the finite case, calculating the multipole expansion of all boxes at all refinement levels. The level-zero expansion then contains the multipole expansion for all particles in the original cube. All of the periodic images have the same multipole expansion about their centers. The fast multipole method requires the local expansion of the potential from all periodic images except the 26 nearest neighbors of the original simulation cell. Having that, the algorithm continues in its downward pass exactly as before.
In the following, we will assume a neutral system of particles. Modifications to include a uniform background to compute systems like the one-component plasma are straightforward, but will not be given here.
Since the multipoles of all the images of the simulation cell are the same, they factor out of the sum over images. The transformation matricesfor each of the reqired simulation cell images can then be added to produce one transformation matrix that, when multiplied times the multipole moments of the simulation cell, produces the coefficients of the localexpansion of all those images.
The algorithm is modified so that step 0 creates the image multipoleto-local transformation described above, and step 3 is modified by adding the nonzero local expansion for levels r = 0 and r = 1. The local expansion for r = 0 in step 3a is simply given by applying this transformation to the multipole moments of the simulation cell. Step 3b does not change in its description, but in the code some logic changes. The changes are actually simplifications. In the finite system, boxes near the faces of the cube are treated differently because there are no neighboring boxes of charge outside
the cube. When periodic boundary conditions are employed, all boxes, whether interior or near the surface, are treated exactly the same. For instance, at level r = 1, where step 3b was not performed in the finite system, one now adds to each of the local expansions of the eight subdivisions of the cube the 189 transformations of the multipole potentials of the subdivisions of the nearby periodic images. Similarly, at all further refinement levels, the full 189 transformations are required for each local expansion. This does not add significantly to the computational requirements of the algorithm, since most of the work occurs at the finest, r = R, level where few of the
boxes are surface boxes.

Answer 1

周期边界条件

我们还制定了一个版本的代码与周期性边界条件评价的潜在能量的一套收费在一个立方体和无限定期图像。通常的程序，涉及埃瓦尔德款项，是有据可查的许多文本。埃瓦尔德款项超过所有无限定期图像的每一个电荷可以进行在一个时间比例为3/ 2。

在快速多极方法，程序的启动，在相同的方式在有限的情况下，计算多极扩大所有箱子都精致程度。该level-zero扩张则包含了多极扩大为所有粒子在原来的立方体。所有的定期图像具有相同的多极展开他们的中心。快速多极方法需要的地方扩大潜在的所有定期图像除了26个近邻的原模拟细胞。有，该算法仍然在其向下通过前完全一样。

在下面，我们将承担一个中性粒子系统。修改包括一个统一的背景来计算系统像单组分血浆是直截了当的，但不会在这里。

由于多极的所有图像的模拟细胞是相同的因素，他们的总和图像。转型中的每一个需求matricesfor模拟细胞图像可以被添加到产生一个变换矩阵，当乘以倍多极矩的模拟细胞，产生系数的localexpansion所有图像。

该算法进行了改进，步骤0创建图像的multipoleto-local了上述，步骤3是通过添加改性的非零膨胀水平=0=1。当地的扩张=0在一步即可应用这个变换的多极矩的模拟细胞。步骤3不改变它的描述，而在代码逻辑的变化。这些变化实际上是简化。在有限的系统，箱子附近的立方体的面是不同的是因为没有相邻的外盒

立方体。当周期边界条件，所有箱子，无论是内部或附近的表面，是完全相同的。例如，在水平=1，3步的地方是在没有进行有限的系统，现在增加了每一个局部扩张的八个分支的立方体的189次变革的多极的潜力的分支附近的定期图像。同样，所有进一步细化水平，全189个转变要求每个地方扩张。这并没有大幅增加的计算要求的算法，因为大多数的工作发生在最好的，=，水平，少数的

箱面框。

这花了我2天啊。。。。

Answer 2

那么多 不是一点点啊。

Answer 3

翻译完有什么好处啊？