Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states

Abstract

International audienceContrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analysis

Mean field limit for bosons with compact kernels interactions by Wigner measures transportation

“…We prove the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in [4,6,11]. …”

“…For instance the method of coherent states [13,15,19] or the BBGKY hierarchy of reduced density matrices ( [7,8,10,9,16,17]), to mention only few. There is a newer approach inspired by ideas from the analysis of oscillation phenomena in finite dimension and relying on Wigner measures (see Definition 1.1 and [3,4,6,5]). With this method the propagation of fairly general quantum states can be understood in the mean field limit ( [4,6]).…”

“…There is a newer approach inspired by ideas from the analysis of oscillation phenomena in finite dimension and relying on Wigner measures (see Definition 1.1 and [3,4,6,5]). With this method the propagation of fairly general quantum states can be understood in the mean field limit ( [4,6]). These quantum states are constrained to a certain compactness property, called the (PI) condition (9), essential for the convergence.…”

“…One expects that loss of compactness would be in fact an obstacle to the achievement of the limit. In [6] the case of a bounded interaction was proven with the specific (PI) condition (9).…”

“…and p > 0 (see [6] for more details). The notations b W ick and P p,q (Z) are explained in Appendix A.3.…”