On the dynamics of correlations in Scaling limits of interacting particle systems

Abstract

In this work, we analyze the properties of two-particle correlations in the weak-coupling and plasma limit of interacting particle systems motivated by Bogolyubov's formal derivation of kinetic equations.<br /> We prove that the leading order evolution in the weak-coupling scaling limit is stable on the macroscopic timescale, and yields the nonlinear Landau equation as kinetic equation. This result shows the transition from the non-Markovian dynamics of the interacting particle system to the Markovian, parabolic evolution in the kinetic limit. Since the system is non-dissipative before taking the limit, we introduce a time-averaged notion of stability to derive an a priori estimate on the solution.<br /> Moreover, we prove the global stability of the truncated correlation function in the plasma limit, for a time independent background distribution and soft potentials. In the case of systems with Coulombian interaction, we prove the onset of the Debye screening length and show that the limit evolution is driven by the interaction of particles with impact parameter much smaller than the Debye length.