Integrability properties and stochastic McKean-Vlasov dynamics with singular Lennard-Jones drift: a mesoscale regularization

Abstract

We study the convergence of the empirical measure of moderately interacting particle systems subject to singular forces derived by Lennard-Jones potential. Although the classical Lennard-Jones force is widely used in molecular dynamics, analytical results are not available. We consider a Lennard-Jones potential with free parameters in the McKean-Vlasov framework and proceed with a regularization at the mesoscale letting the particles interact moderately. We prove the well-posedness of the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards the solution of the McKean-Vlasov Fokker-Planck PDE, by means of a semigroup approach. We derive both the range of parameters characterizing the aggregation and repulsive force and the mesoscale order for which the convergence is achieved, by obtaining the right integrability regularity of the drift.