A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics

TL;DR

This paper presents a novel method for deriving fermionic mean-field dynamics from microscopic quantum equations, extending previous bosonic approaches, and applies it to systems with long-range and Coulomb interactions, providing explicit convergence rates.

Contribution

It adapts a bosonic mean-field derivation method to fermionic systems, handling long-range and Coulomb interactions with explicit convergence rates.

Findings

Derived fermionic mean-field equations from microscopic dynamics.

Handled long-range and Coulomb interactions with regularity assumptions.

Provided explicit convergence rates for initial data close to antisymmetrized product states.

Abstract

We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schroedinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in [Pickl, Lett. Math. Phys., 97(2):151-164, 2011] for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.