A Remark on the Mean-Field Dynamics of Many-Body Bosonic Systems with Random Interactions and in a Random Potential

Abstract

The mean-field limit for the dynamics of bosons with random two-body interactions and in the presence of a random external potential is rigorously studied, both for the Hartree dynamics and the Gross–Pitaevskii dynamics. First, it is shown that, for interactions and potentials that are almost surely bounded, the many-body quantum evolution can be replaced in the mean-field limit by a single particle nonlinear evolution that is described by the Hartree equation. This is an Egorov-type theorem for many-body quantum systems with random interactions. The analysis is then extended to derive the Gross–Pitaevskii equation with random interactions.