On the derivation of the Hartree equation in the mean field limit:   Uniformity in the Planck constant
  In this paper the Hartree equation is derived from the $N$-body Schr\''odinger equation in the mean-field limit, with convergence rate estimates that are uniform in the Planck constant $\hbar$. Specifically, we consider the two following cases:(a) T\''oplitz initial data and Lipschitz interaction forces, and (b) analytic initial data and interaction potential, over short time intervals independent of $\hbar$. The convergence rates in these two cases are $1/\sqrt{\log\log N}$ and $1/N$ respectively. The treatment of the second case is entirely self-contained and all the constants appearing in the final estimate are explicit. It provides a derivation of the Vlasov equation from the $N$-body classical dynamics using BBGKY hierarchies instead of empirical measures. 
Fran\c{c}ois Golse (CMLS), Thierry Paul (CMLS), Mario Pulvirenti, Ois   Fran\c{c}, Thierry Golse, Mario Paul
math.AP math-ph math.MP quant-ph