Some ideas about quantitative convergence of collision models to their mean field limit

We consider a stochastic $N$-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when $N\to \infty$. For any time $T>0$ we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in some homogeneous negative Sobolev space. The control we get is Gaussian, i.e. we prove that the distance is bigger than $x N^{-1/2}$ with a probability of type $O(e^{-x^2})$. The two main ingredients are first a control of fluctuations due to the discrete nature of collisions, secondly a Lipschitz continuity for the Boltzmann collision kernel. The latter condition, in our present setting, is only satisfied for Maxwellian models. Numerical computations tend to show that our results are useful in practice.Comment: 27 pages, references added and style improve

Amniotic fluid epidermal growth factor levels in normal and abnormal pregnancies

OBJECTIVE: TO determine the concentrations of epidermal growth factor (EGF) in amniotic fluid (AF) from women during late pregnancy, with and without pathophysiologic complications