The rigorous derivation of Vlasov equations with local alignments from moderately interacting particle systems

In this paper, we present a rigorous derivation of the mean-field limit for a moderately interacting particle system in $\R^d$ $(d\geq 2)$. For stochastic initial data, we demonstrate that the solution to the interacting particle model, with an appropriately applied cut-off, converges in probabilistic sense to the solution of the characteristics of the regularized Vlasov models featuring local alignments and Newtonian potential. Notably, the cutoff parameter for the singular potential is selected to scale polynomially with the number of particles, representing an improvement over the logarithmic cut-off obtained in [38].