Abstract
We provide a rigorous derivation of Einstein’s formula for the effective viscosity of dilute suspensions of n rigid balls, set in a volume of size 1. So far, most justifications were carried under a strong assumption on the minimal distance between the balls:
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 modeled by standard Poisson processes with almost minimal hardcore condition.
Notes
1 In detail: let be the operator that erases all points without a neighboring point closer than η, and let Tx denote a translation by x. Now, let μ be the measure for the original process
Then the measure for
is given by
Since
(for all x, in particular for
), we have for any measurable set A that
This immediately implies that the new process adopts stationarity and ergodicity.