From Particle Systems to the Landau Equation: A Consistency Result

We consider a system of N classical particles, interacting via a smooth, short-range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in terms of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N →∞, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.

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