Mean field limit of bosonic systems in partially factorized states and their linear combinations

TL;DR

This paper investigates the mean field limit of bosonic systems with partially factorized initial states and their superpositions, demonstrating convergence of reduced density matrices to either pure or mixed states depending on the initial configuration.

Contribution

It introduces a rigorous analysis of the mean field limit for bosonic systems starting from partially factorized states and their linear combinations, extending previous results to more general initial conditions.

Findings

Reduced density matrices converge to pure states in trace norm for single initial states.

Convergence to mixed states in Hilbert-Schmidt norm for linear superpositions.

Results apply to systems with a fixed number of particles in initial states.

Abstract

We study the mean field limit of one-particle reduced density matrices, for a bosonic system in an initial state with a fixed number of particles, only a fraction of which occupies the same state, and for linear combinations of such states. In the mean field limit, the time-evolved reduced density matrix is proved to converge: in trace norm, towards a rank one projection (on the state solution of Hartree equation) for a single state; in Hilbert-Schmidt norm towards a mixed state, combination of projections on different solutions (corresponding to each initial datum), for states that are a linear superposition.