Derivation of the Boltzmann equation with moderately soft potentials from a perturbed Nanbu particles system

We derive the 3D spatially homogeneous Boltzmann's equation with moderately soft potentials and singular angular interaction, from an interacting particles system. The collision kernel is of the form $B(z,\sigma)=|z|^{\gamma}b\left( \frac{z}{|z|}\cdot \sigma\right)$ and for $K>0$, $\sin(\theta)b\left(\cos(\theta)\right)\sim K\theta^{-1-\nu}$, with $\gamma\in (-2,-1)$ and $\nu\in(1,2)$ satisfying $\gamma+\nu>0$. We use at the particle level the regularizing effects of the grazing collisions, in order to control the singularity of the soft potential. This enables to use a classical compactness argument, and provide a qualitative convergence result from the interacting particles system toward the solution of the limit macroscopic equation.