The Boltzmann–grad limit of a hard sphere system. Analysis of the correlation error

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, the correlation errors, measuring the deviations in time from the statistical independence of particles (propaga- tion 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 < ε−α, such an error flows to zero with the average density ε, for short times, as εγk, for some positive α,γ ∈ (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 ε. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many–recollision events.