Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
May 23-26, 2011

Van Vleck Hall
Department of Mathematics
University of Wisconsin -Madison
Madison, WI 53706

Second order correction to the mean field of weakly interacting Bosons

Prof. Manoussos Grillakis

University of Maryland

Abstract:   In the present talk I would like to consider the evolution of N identical quantum particles (Bosons), for N large but finite. The basic problem is that of an effective, approximate, description of the evolution. The mean field approximation achieves this goal by describing all particles by the same wavefunction F (t, x) which satisfies the Hartree evolution equation. However the approximation is only in an average sense and valid only for times of the order log(N). I will describe a second order correction in terms of a kernel function k (t, x, y) which describes fluctuations from the mean field and derive a Schrödinger type evolution equation for k which is coupled with the Hartree dynamics. The correction tracks the exact dynamics for times of the order vN. This work is in collaboration with Matei Machedon and Dio Margetis.