Browsing by Author "Petrat, S"

Now showing 1 - 6 of 6
  • Thumbnail Image
    ItemOpen Access
    A New Method and a New Scaling for Deriving Fermionic Mean-Field Dynamics
    (Mathematical Physics, Analysis and Geometry, 2016-03) Petrat, S; Pickl, P
    © 2016, The Author(s). 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.
  • Thumbnail Image
    ItemOpen Access
    Bogoliubov corrections and trace norm convergence for the Hartree dynamics
    Mitrouskas, D; Petrat, S; Pickl, P
    We consider the dynamics of a large number N of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space that is very similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the N-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for general initial states, e.g., those that are expected to be ground states of interacting systems.
  • Thumbnail Image
    ItemOpen Access
    Derivation of the Bogoliubov Time Evolution for Gases with Finite Speed of Sound
    Petrat, S; Pickl, P; Soffer, A
    The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, which lead to a much more precise description of both the ground state properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically allow a convergence result for the reduced density matrix only, one gets norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In the present paper we consider an interacting Bose gas in the ground state with slight perturbations. We consider the case where the volume of the gas - in units of the support of the excitation - and the density of the gas tend to infinity simultaneously. We assume that the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (non-zero) speed of sound in the gas. We show that the difference between the N-body description and the Bogoliubov description is small in $L^2$ as the density of the gas tends to infinity. In this situation the ratio of the occupation number of the ground-state and the excitation forming the fluctuations will influence the leading order of the dynamics of the system. In this sense we show the validity of the Bogoliubov time evolution in a situation where the temperature has an effect on the dynamics of the system.
  • Thumbnail Image
    ItemOpen Access
    Effective dynamics of a tracer particle in a dense homogeneous quantum gas
    Jeblick, M; Mitrouskas, D; Petrat, S; Pickl, P
    We investigate the mean field regime of the dynamics of a tracer particle in a homogenous quantum gas. For a bosonic gas, we show that this regime is constrained by the well known requirement of an appropriate mean field scaling of the interaction. For fermions, however, we find an important qualitative difference. Not only are fermions much more homogeneously distributed than bosons but also deviations from the mean are due only to fast degrees of freedom in the gas. This observation leads to an explanation of why a tracer particle behaves freely in the dense homogeneous fermion gas despite of a non-scaled interaction, i.e., despite of non-vanishing statistical fluctuations. Finally, we indicate how the gained insight can be rigorously justified.
  • Thumbnail Image
    ItemOpen Access
    Free Time Evolution of a Tracer Particle Coupled to a Fermi Gas in the High-Density Limit
    (Communications in Mathematical Physics, 2017-11) Jeblick, M; Mitrouskas, D; Petrat, S; Pickl, P
    © 2017, Springer-Verlag GmbH Germany. The dynamics of a particle coupled to a dense and homogeneous ideal Fermi gas in two spatial dimensions is studied. We analyze the model for coupling parameter g = 1 (i.e., not in the weak coupling regime), and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift which is given at leading order by the spatially homogeneous mean field potential of the gas particles. Here, the mean field approximation turns out to be accurate although the fluctuations of the potential around its mean value can be arbitrarily large. Our result is in contrast to a dense bosonic gas in which the free motion of a tracer particle would be disturbed already on a very short time scale. The proof is based on the use of strong phase cancellations in the deviations of the microscopic dynamics from the mean field time evolution.
  • Thumbnail Image
    ItemOpen Access
    Kinetic energy estimates for the accuracy of the time-dependent Hartree–Fock approximation with Coulomb interaction
    (Journal de Mathématiques Pures et Appliquées, 2016-01) Bach, V; Breteaux, S; Petrat, S; Pickl, P; Tzaneteas, T
    © 2015 The Authors. We study the time evolution of a system of N spinless fermions in R3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.