A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics

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 Schrödinger 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 meanfield 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 Sören Petrat

Mean-field evolution of fermionic systems

“…Recently, this resut has been generalized by Benedikter-Porta-Schlein [5] to a much larger class of interactions (essentially, twice differentiable); convergence was proven for all times, with effective estimates on the rate of convergence. Later, the same result has been obtained by Petrat-Pickl [22], with a different method. The result of [5] applies to initial data describing pure states (e.g., approximate Slater determinants); very recently a similar result has been proven for mixed states (describing positive temperature states) by Benedikter-Jaksic-Porta-Saffirio-Schlein [7].…”

“…Finally, a regularized Coulomb interaction with a N −2/3 coupling constant has been studied by Petrat-Pickl [22].…”

Mean-Field Dynamics for the Nelson Model with Fermions

“…To prove Theorem II.3 it is important to define and control the right macroscopic variables. For that, we adapt techniques that are based on the method from [36] and that were further developed in [34,29,30]. In addition, it is crucial to find the right measure for the correlations between the electrons and to consider only initial states with semiclassical structure.…”

“…Before we prove appropriate estimates for the time derivative of β(t), let us state a technical lemma which was already proven, e.g., in [34,3]; we give a proof here for convenience. Note that this is an important point where the antisymmetry of the wave function is used.…”

Semi-classical limit of large fermionic systems at positive temperature

“…For atoms the positive Thomas-Fermi model was derived for the first time in [43]. There are several mathematical works on the time-dependent setting [42,52,3,14,17,8,5,1,44,7,13], in which the Schrödinger dynamics has been proved to converge to the Vlasov time-dependent equation in the limit N → ∞. Finally, the first two terms in the expansion of the (free) energy of a Fermi gas with spin in the limit ρ → 0 was provided in [32] at T = 0 and in [50] at T > 0.…”