We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of $n$ rigid balls, $n \gg 1$, set in a volume of size $1$. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{min} \ge c n^{-\frac{1}{3}}$, $c > 0$. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition.

中文翻译：

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爱因斯坦有效粘度公式推导的温和假设

我们提供了爱因斯坦公式的严格推导，用于 $n$ 刚性球的稀释悬浮液的有效粘度，$n \gg 1$，设置在大小为 $1$ 的体积中。到目前为止，大多数理由都是在球之间最小距离的强烈假设下进行的：$d_{min} \ge cn^{-\frac{1}{3}}$, $c > 0$。我们将这个假设放宽到一组两个弱得多的条件：一个基本上表示球不重叠，而另一个则控制彼此靠近的球的数量。特别是，我们的分析涵盖了由标准泊松过程建模的悬架案例，几乎没有硬核条件。