  We propose a deterministic particle method for a one-dimensional nonlocal
equation with interactions through the repulsive Morse potential. We show that
the particle method converges as the number of particles goes to infinity
towards weak measure solutions to the nonlocal equation. Such a results is
proven under the assumption of initial data in the space of probability
measures with finite second moment. In particular, our method is able to
capture a measure-to-$L^\infty$ smoothing effect of the limit equation.
Moreover, as the Morse potential is rescaled to approach a Dirac delta,
corresponding to strongly localised repulsive interactions, the scheme becomes
a particle approximation for the quadratic porous medium equation. We show that
in the joint limit (localised repulsion and increasing number of particles) the
reconstructed density converges to a weak solution of the porous medium
equation. The strategy relies on various estimates performed at the particle
level, including $L^p$ estimates and an entropy dissipation estimate, which
benefit from the particular structure of our particle scheme and from the
absolutely continuous reconstruction of the density from the particle
locations.
