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

We give a rigorous derivation of the Hartree equation for the many-body dynamics of pseudo-relativistic Fermi systems at high density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ϱ</mml:mi> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , on arbitrarily large domains, at zero temperature. With respect to previous works, we show that the many-body evolution can be approximated by the Hartree dynamics locally, proving convergence of the expectation of observables that are supported in regions with fixed volume, independent of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϱ</mml:mi></mml:math> . The result applies to initial data describing fermionic systems at equilibrium confined in arbitrarily large domains, under the assumption that a suitable local Weyl-type estimate holds true. The proof relies on the approximation of the initial data through positive temperature quasi-free states, that satisfy strong local semiclassical bounds, which play a key role in controlling the growth of the local excitations of the quasi-free state along the many-body dynamics.