94 research outputs found
A New Method and a New Scaling For Deriving Fermionic Mean-field Dynamics
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.Comment: 42 pages, LaTex; v2: introduction expanded, presentation of main
results improved, several minor improvements and references adde
Mean-field limits of particles in interaction with quantized radiation fields
We report on a simple strategy to treat mean-field limits of quantum
mechanical systems in which a large number of particles weakly couple to a
second-quantized radiation field. Extending the method of counting, introduced
in [Lett. Math. Phys. 97, 151-164], with ideas inspired by
[http://www.mathematik.uni-muenchen.de/%7Ebohmmech/theses/Matulevicius_Vytautas_MA.pdf]
and [J. Math. Phys. 54(1), 012303] leads to a technique that can be seen as a
combination of the method of counting and the coherent state approach. It is
similar to the coherent state approach but might be slightly better suited to
systems in which a fixed number of particles couple to radiation. The strategy
is effective and provides explicit error bounds. As an instructional example we
derive the Schr\"odinger-Klein-Gordon system of equations from the Nelson model
with ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate of
convergence of the one-particle reduced density matrix of the non-relativistic
particles in Sobolev norm. More complicated models like the Pauli-Fierz
Hamiltonian can be treated in a similar manner [arXiv:1609.01545].Comment: A final version (improved presentation and many more references) will
appear in "Macroscopic Limits of Quantum Systems", Springer Proceedings in
Mathematics and Statistic
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