Propagation of Chaos for 2D Log Gas on the Whole Space

We derive the quantitative propagation of chaos in the sense of relative entropy for the first time for the 2D Log gas or the weakly interacting particle systems with 2D Coulomb interactions on the whole space. We resolve this problem by adapting the modulated free energy method in [BJW23] to the whole space setting and establishing the crucial logarithmic growth estimates for the mean-field Poisson-Nernst-Planck (PNP) equation of single component via the parabolic maximum principle.