Fluctuations Around Hartree Dynamics

Benjamin Schlein

We report on recent results concerning the derivation of e ective evolution equations starting from many body quantum dynamics. In particular, we obtain rigorous derivations of nonlinear Hartree equations in the bosonic mean eld limit, with precise bounds on the rate of convergence. Moreover, we present a central limit theorem for the fluctuations around the Hartree dynamics.

Li Chen, Ji Oon Lee, Benjamin Schlein

Simon Buchholz, Chiara Saffirio, Benjamin Schlein

We consider the time evolution of N bosons in the mean field regime for factorized initial data. In the limit of large N, the many body evolution can be approximated by the non-linear Hartree equation. In this paper we are interested in the fluctuations around the Hartree dynamics. We choose k self-adjoint one-particle operators O 1,…,O k on $L^{2} ({\mathbb{R}}^{3})$ , and we average their action over the N-particles. We show that, for every fixed $t \in{\mathbb{R}}$ , expectations of products of functions of the averaged observables approach, as N→∞, expectations with respect to a complex Gaussian measure, whose covariance matrix can be expressed in terms of a Bogoliubov transformation describing the dynamics of quantum fluctuations around the mean field Hartree evolution. If the operators O 1,…,O k commute, the Gaussian measure is real and positive, and we recover a "classical" multivariate central limit theorem. All our results give explicit bounds on the rate of the convergence.

Mathieu Lewin B, Phan Thành Nam, Benjamin Schlein

We consider the dynamics of a large system of $N$ interacting bosons in the mean-field regime where the interaction is of order $1/N$. We prove that the fluctuations around the nonlinear Hartree state are generated by an effective quadratic Hamiltonian in Fock space, which is derived from Bogoliubov's approximation. We use a direct method in the $N$-particle space, which is different from the one based on coherent states in Fock space.

M. Sallé, Jan Smit A, Jeroen C. Vink