Consider two unit balls in a $d$-dimensional flat torus with edge length $r$, for $dgeq 2$. The balls do not move by themselves but they are pushed by a Brownian motion. The balls never intersect---they reflect if they touch. It is proved that the joint distribution of the processes representing the centers of the balls converges to the distribution of two independent Brownian motions when $ro infty$, assuming that we use a proper clock and proper scaling. The diffusion coefficient of the limit process depends on the dimension. The positions of the balls are asymptotically independent also in the following sense. The rescaled stationary distributions of the centers of the balls converge to the product of the stationary (hence uniform) distributions for each ball separately, as $roinfty$.
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机译：考虑一个$ d $维扁平圆环中的两个单位球，边缘长度为$ r $，为$ d geq 2 $。球本身不会移动，但是会受到布朗运动的推动。球永远不会相交-如果它们接触，它们就会反射。证明了，假设我们使用适当的时钟和适当的缩放比例，表示球中心的过程的联合分布收敛到两个独立的布朗运动的分布。极限过程的扩散系数取决于尺寸。球的位置在以下意义上也是渐近独立的。球中心的重新缩放后的静态分布收敛为每个球的静态（因此均匀）分布的乘积，分别为$ r to infty $。
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