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

Sublattice identification in scanning force microscopy on alkali halide surfaces

We propose and apply to the KBr(001) surface a new procedure for species recognition in scanning force microscopy (SFM) of ionic crystal surfaces which show a high symmetry of the charge arrangement. The method is based on a comparison between atomistic simulations and site-specific frequency versus distance measurements. First, by taking the difference of force-distance curves extracted at a few judiciously chosen surface sites we eliminate site-independent long-range forces. The obtained short-range force differences are then compared with calculated ones assuming plausible tip apex models. This procedure allows for the first time identification of the tip apex polarity and of the positive and negative sublattices in SFM images of the (001) cleavage surface of an ionic crystal with the rock salt structure