Smoluchowski-Kramers Approximation In Mean Field Limit Of Stochastic Particle System

This thesis is concerned with both the small mass limit and the mean field limit for a stochastic N-particle system with a small mass ?.Particles in the system are forced by the friction which is proportional to the velocity,and random fluctuation which is described by white noise,also there are interaction between particles.We show the limit of the empirical measure associated with the solution of the stochastic particle system as both the number of particles N?? and the mass ??0.The result can be seen as the Smoluchowski-Kramers approximation in the mean field limit.To do this we first establish a Smoluchowski-Kramers approximation result for the corresponding mean field characteristic equation(which is a distribution dependent stochastic differential equation)of the system.Then we have the corresponding mean field PDE.Further the approximation relation between the solution of the particle system and the solution of the mean field characteristic equation as N?? yields the limit of the empirical measure.Finally,the result also implies a result about the singular limit of the mean field PDEs.