  We present a quantitative analysis of the Boltzmann-Grad (low-density) limit
of a hard sphere system. We introduce and study a set of functions (correlation
errors) measuring the deviations in time from the statistical independence of
particles (propagation of chaos). In the context of the BBGKY hierarchy, a
correlation error of order $k$ measures the event where $k$ particles are
connected by a chain of interactions preventing the factorization. We show
that, provided $k < \varepsilon^{-\alpha}$, such an error flows to zero with
the average density $\varepsilon$, for short times, as $\varepsilon^{\gamma
k}$, for some positive $\alpha,\gamma \in (0,1)$. This provides an information
on the size of chaos, namely, $j$ different particles behave as dictated by the
Boltzmann equation even when $j$ diverges as a negative power of $\varepsilon$.
The result requires a rearrangement of Lanford perturbative series into a
cumulant type expansion, and an analysis of many-recollision events.
