Propagation of chaos for large Brownian particle system with Coulomb interaction

We investigate a system of N Brownian particles with the Coulomb interaction in any dimension d≥ 2 , and we assume that the initial data are independent and identically distributed with a common density ρ0 satisfying ∫Rdρ0lnρ0dx<∞ and ρ0∈L2dd+2(Rd)∩L1(Rd,(1+|x|2)dx). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For d= 2 , we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When N→ ∞, the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.

Duke Scholars

Author Jian-Guo Liu Physics