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arXiv:2203.11255 · Journal of Mathematical Physics

Effective dynamics of interacting fermions from semiclassical theory to the random phase approximation

Niels Benedikter

Published Aug 1, 2022• 10 citations
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Abstract

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 semiclassical theory given by the Vlasov equation, the mean-field theory given by the Hartree–Fock equation, and the description of the dominant effects of non-trivial entanglement by the random phase approximation. Particular attention is given to the discussion of admissible initial data, and I present an example of a realistic quantum quench that can be approximated by Hartree–Fock dynamics.

DOI
10.1063/5.0091694

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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 semiclassical theory given by the Vlasov equation, the mean-field theory given by the Hartree–Fock equation, and the description of the dominant effects of non-trivial entanglement by the random phase approximation. Particular attention is given to the discussion of admissible initial data, and I present an example of a realistic quantum quench that can be approximated by Hartree–Fock dynamics.
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 fermions interacting via inverse power law potentials in the mean-field and semiclassical regimes, reviewing the material presented at the 11th summer school "Methods and Models of Kinetic Theory" held in Pesaro in June 2022. More precisely, we focus on weakly interacting fermions whose collective effect can be approximated by an averaged potential in convolution form, and review recent mean-field techniques based on second quantization approaches. As a first step we obtain a reduced description given by the time-dependent Hartree-Fock equation. As a second step we look at longer time scales where a semiclassical description starts to be relevant and approximate the many-body dynamics with the Vlasov equation, which describes the evolution of the effective probability density of particles on the one particle phase space.
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 semiclassical limit. Assuming some regularity of the interaction potential we show that the many body evolution of initial states close to Slater determinants exhibiting the appropriate semiclassical structure can be approximated by the Hartree-Fock equation. Our method provides precise bounds on the rate of the convergence.
We study the dynamics of many-body Fermi systems, for a class of initial data which are close to quasi-free states exhibiting a nonvanishing pairing matrix. We focus on the mean-field scaling, which for fermionic systems is naturally coupled with a semiclassical scaling. Under the assumption that the initial datum enjoys a suitable semiclassical structure, we give a rigorous derivation of the time-dependent Hartree-Fock-Bogoliubov equation, a nonlinear effective evolution equation for the generalized one-particle density matrix of the system, as the number of particles goes to infinity. Our result holds for all macroscopic times, and provides bounds for the rate of convergence.
In these notes, based on a mini-course held at the summer school "Methods and Models of Kinetic Theory" that took place in Porto Ercole in June 2016, we review some of the recent developments in the derivation of effective evolution equations starting from many-body quantum mechanics. We discuss the derivation of the Hartree equation in the bosonic mean-field limit, of the Gross-Pitaevskii equation describing the dynamics of initially trapped Bose-Einstein condensates and of the Hartree-Fock equation for fermions in a joint mean-field and semiclassical limit.
In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.