Discrete-to-continuum limits of interacting particle systems in one dimension with collisions

We study a class of interacting particle systems in which n signed particles move on the real line. This class includes the overdamped dynamics (i.e. 'velocity = force') of charged particles interacting by the Coulomb potential. Particles of opposite sign collide typically in finite time. Upon collision, pairs of colliding particles are removed from the system. In a recent paper by Peletier, Požár and the author, the specific case of the Coulomb potential was studied. Global well-posedness of the corresponding particle system was shown and a discrete-to-continuum limit (i.e. n→∞) to a nonlocal PDE for the signed particle density was established. Both results rely on innovative use of techniques in ODE theory and viscosity solutions. The particle system with the Coulomb potential has a limited range of applications. The aim of the present paper is to broadly widen this range by adding an external potential and by considering a large class of interaction potentials which, moreover, may scale with n. Several key steps in the proofs for the Coulomb potential do not carry over; new techniques are developed to establish these steps.