Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Abstract

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N (x i − x j )), where x = (x 1 , . . . , x N ) denotes the positions of the particles. Let H N denote the Hamiltonian of the system and let ψ N,t be the solution to the Schrödinger equation. Suppose that the initial data ψ N,0 satisfies the energy condition. .. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψ N,t are also asymptotically factorized and the one particle orbital wave function solves the GrossPitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential V . We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψ N,0 is assumed in a stronger sense.

Kac’s program in kinetic theory

“…The so-called BBGKY hierarchy method (Bogoliubov, Born, Green, Kirkwood and Yvon) is very popular in physics and mathematics for studying many-particle systems: see for instance [3] where this approach is used for Kac's master equation for hard spheres, or see, among many other works, the recent series of papers [4,25,26,27] where this approach is used for the derivation of nonlinear mean-field Schrödinger equations in quantum physics. The basic ideas underlying this approach to mean-field limit could be summarized as:…”

New approach to derivation of quantum kinetic equations with initial correlations

“…In case of pure states the quantum Vlasov-type kinetic equation with initial correlations (26) can be reduced to the Gross-Pitaevskii-type kinetic equation [14]. Indeed, in this case the one-particle density operator f 1 (t) = |ψ t ψ t | is a one-dimensional projector onto a unit vector |ψ t ∈ H and its kernel has the following form: f 1 (t, q, q ′ ) = ψ(t, q)ψ * (t, q ′ ).…”

“…The conventional approach to the rigorous derivation of the quantum kinetic equations is based on the consideration of an asymptotic behavior of a solution of the quantum BBGKY hierarchy for marginal density operators constructed within the framework of the theory of perturbations in case of initial states specified by a one-particle (marginal) density operator without correlations [11][12][13][14], i.e. such that satisfy a chaos condition.…”

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

“…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]). 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.…”