Quantum Dynamics with Mean Field Interactions: a New Approach

TL;DR

This paper introduces a novel method for analyzing the time evolution of bosonic systems under mean-field interactions, providing quantitative bounds on how the many-particle dynamics approximates the Hartree equation.

Contribution

A new technique controlling correlation growth in bosonic systems, yielding explicit convergence rates between many-particle and mean-field dynamics.

Findings

Convergence of the one-particle density matrix at rate 1/N.

Quantitative bounds on the difference between many-particle and Hartree dynamics.

Applicability to bounded interaction potentials.

Abstract

We propose a new approach for the study of the time evolution of a factorized $N$-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schr\"odinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the $N$-particle Schr\"odinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order 1/N for all fixed times.