  We prove a strong/weak stability estimate for the 3D homogeneous Boltzmann
equation with moderately soft potentials ($\gamma\in(-1,0)$) using the
Wasserstein distance with quadratic cost. This in particular implies the
uniqueness in the class of all weak solutions, assuming only that the initial
condition has a finite entropy and a finite moment of sufficiently high order.
We also consider the Nanbu $N$-stochastic particle system which approximates
the weak solution. We use a probabilistic coupling method and give, under
suitable assumptions on the initial condition, a rate of convergence of the
empirical measure of the particle system to the solution of the Boltzmann
equation for this singular interaction.
