相关论文: From the Hartree dynamics to the Vlasov equation

We consider the evolution of $N$ fermions interacting through a Coulomb or gravitational potential in the mean-field limit as governed by the nonlinear Hartree equation with Coulomb or gravitational interaction. In the limit of large $N$,…

We consider mixed quasi-free states describing $N$ fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large $N$ limit, we study the convergence towards the classical…

We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the…

We investigate the semiclassical limit from the semi-relativistic Hartree-Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular…

We review recent results concerning the evolution of fermionic systems. We are interested in the mean field regime, where particles experience many weak collisions. For fermions, the mean field regime is naturally linked with a…

We consider the semiclassical limit from the Hartree to the Vlasov equation with general singular interaction potential including the Coulomb and gravitational interactions, and we prove explicit bounds in the strong topologies of Schatten…

The mean field limit for systems of many fermions is naturally coupled with a semiclassical limit. This makes the analysis of the mean field regime much more involved, compared with bosonic systems. In this paper, we study the dynamics of…

We consider the quantum dynamics of $N$ interacting fermions in the large $N$ limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect…

We obtain the combined mean-field and semiclassical limit from the $N$-body Schr\"{o}dinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation…

We review the recent results [45, 46] concerning the semiclassical limit from the Hartree dynamics to the Vlasov equation with singular potentials and extend them to the case of more general radial interactions. We prove that, at positive…

In this work, we derive the time-dependent Hartree(-Fock) equations as an effective dynamics for fermionic many-particle systems. Our main results are the first for a quantum mechanical mean-field dynamics for fermions; in previous works,…

We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential $V(x)=1/|x|^{\alpha}$, for $\alpha\in(0,1]$. We prove the convergence of a solution of the many-body Schr\"{o}dinger equation to a…

I review results concerning the derivation of effective equations for the dynamics of interacting Fermi gases in a high-density regime of mean-field type. Three levels of effective theories, increasing in precision, can be distinguished:…

The derivation of effective macroscopic theories approximating microscopic systems of interacting particles is a major question in non-equilibrium statistical mechanics. In these notes we present an approximation of systems made by many…

We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and…

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution…

We study the dynamics of a Fermi gas with a Coulomb interaction potential, and show that, in a mean-field limiting regime, the dynamics is described by the Hartree-Fock equation. This extends previous work of Bardos et al. to the case of…

We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which…

In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the $N$-body linear Schr\"{o}dinger equation uniformly in N leading to the N-body Liouville equation of…

We study the dynamics of two-dimensional interacting fermions submitted to a homogeneous transverse magnetic field. We consider a large magnetic field regime, with the gap between Landau levels set to the same order as that of potential…