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Communications in Mathematical Sciences

The TDHF Approximation for Hamiltonians with mmm-particle Interaction Potentials

Claude Bardos, Bernard Ducomet A, François Golse, Alex D. Gottlieb, Norbert J. Mauser

Published Jan 1, 2007• 14 citations
Publisher

Abstract

According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N .In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity.In this article we extend our results to systems of fermions with m-particle interactions (m > 2).

DOI
10.4310/cms.2007.v5.n5.a2

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According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N .In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity.In this article we extend our results to systems of fermions with m-particle interactions (m > 2).
This article examines the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find the TDHF approximation to be accurate when there are sufficiently many particles and the initial many-particle state is a Slater determinant, or any Gibbs equilibrium state for noninteracting fermions. Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the the TDHF approximation vanishes at all times in the mean field limit.
This article concerns the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find that the TDHF approximation is accurate when there are sufficiently many particles and the initial many-particle state is any Gibbs equilibrium state for noninteracting fermions (with Slater determinants as a special example). Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the TDHF approximation vanishes at all times in the mean field limit.
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 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.
The time-dependent Hartree and Hartree-Fock equations provide effective mean-field descriptions for the dynamics of large fermionic systems and play a fundamental role in many areas of physics. In this work, we rigorously derive the time-dependent Hartree equations as the large-NNN limit of the microscopic Schrödinger dynamics of NNN fermions confined to a volume of order one and interacting via strong pair potentials. A central step in our analysis is the implementation of time-dependent gauge transformations, which eliminate the dominant contribution from the interaction potential in both the Schrödinger and Hartree evolutions.
We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].