MCQM22 Third School and Workshop

Mathematical Challenges in Quantum Mechanics

Como, Italy -- June 13 - 18, 2022

Marcello Porta

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

Abstract: I will discuss the evolution of many-body Fermi gases in three dimensions, in large domains. In many physically relevant applications, due to the large number of particles involved, it is essentially impossible to extract quantitative information on the dynamics from the solution of the Schroedinger equation. However, in suitable scaling regimes one expects that the dynamics of local observables can be effectively studied via nonlinear evolution equations, which involve much less degrees of freedom, and where the effect of all particles on a single one is taken into account by a self-consistent potential. A famous example of such non-linear dynamics is the Hartree-Fock equation, for the evolution of many-body Fermi gases. In the last years, there has been important progress in the derivation of the Hartree-Fock equation from the Schroedinger equation, in the mean-field regime. The main limitation is that this regime does not allow to study extended systems: the initial datum is prepared in a volume of order 1, which means that the density is of the order of the total number of particles, and the interaction is scaled with the inverse of the number of particles.

In this talk I will discuss the dynamics of many-body Fermi gases, in a joint semiclassical/high density regime, in arbitrarily large domains. This setting can be viewed as a Kac limit for interacting fermions. For a class of initial data describing zero-temperature states I will prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, with a rate of convergence that is independent of the volume of the system. The result holds provided a suitable non-concentration estimate for the density of the system holds true, which we establish for short macroscopic times. The result is the natural extension of previous work on the mean-field regime to the setting of Fermi gases in the thermodynamic limit.

Joint work with Luca Fresta (University of Bonn) and Benjamin Schlein (University of Zürich).