Derivation of Mean-field Dynamics for Fermions

Abstract

In this work, we derive the time-dependent Hartree(-Fock) equations as an effective dynamics for fermionic many-particle systems. Our main results are the first for a quantum mechanical mean-field dynamics for fermions; in previous works, the mean-field limit is usually either coupled to a semiclassical limit, or the interaction is scaled down so much, that the system behaves freely for large particle number $N$. We consider the fermionic Hartree equations (i.e., the Hartree-Fock equations without exchange term) in this work, since the exchange term is subleading in our setting. The main result is that the fermionic Hartree dynamics approximates the Schrödinger dynamics well for large $N$. We give explicit values for the rates of convergence. We prove two types of results. The first type is very general and concerns arbitrary free Hamiltonians (e.g., relativistic, non-relativistic, with external fields) and arbitrary interactions. The theorems give explicit conditions on the solutions to the fermonic Hartree equations under which a derivation of the mean-field dynamics succeeds. The second type of results scrutinizes situations where the conditions are fulfilled. These results are about non-relativistic free Hamiltonians with external fields, systems with total kinetic energy bounded by $const \cdot N$ and with long-range interactions of the form $|x|^{-s}$, with $0 < s < \frac{6}{5}$ (sometimes, for technical reasons, with a weaker or cut off singularity). We prove our main results by using a new method for deriving mean-field dynamics developed by Pickl in [Lett. Math. Phys., 97(2):151-164, 2011]. This method has been applied successfully in quantum mechanics for deriving the bosonic Hartree and Gross-Pitaevskii equations. Its application to fermions in this work is new. Finally, we show how also the recently treated semiclassical mean-field limits can be derived with this method.