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.
Highlights
The behavior of an interacting many-body system in classical or quantum mechanics can be very complicated and the microscopic equations governing its behavior are usually practically impossible to solve for more than three or four particles
One type of such a statistical description is to approximate the microscopic dynamics by an effective one-body dynamics, i.e., replace the microscopic evolution equation with very many degrees of freedom by a simpler, usually non-linear equation with very few degrees of freedom
Famous examples are the Boltzmann, Navier-Stokes and Vlasov equations in classical mechanics, and the Hartree, Hartree-Fock and GrossPitaevskii equations in quantum mechanics; different regimes can lead to different effective evolution equations, e.g., kinetic equations or mean-field equations
Summary
Math Phys Anal Geom (2016)19:3 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. Keywords Mean-field limit · Fermionic mean-field dynamics · Reduced Hartree-Fock · Many-body quantum mechanics
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