The semi-classical limit of large fermionic systems

Søren Fournais (1) , Mathieu Lewin (2) , Jan Philip Solovej (3)

1 Department of Mathematical [Aarhus]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Mathematical Sciences [Copenhagen]

We study a system of $N$ fermions in the regime where the intensity of the interaction scales as $1/N$ and with an effective semi-classical parameter $\hbar=N^{-1/d}$ where $d$ is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas-Fermi minimizers in the limit $N\to\infty$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti-Hewitt-Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Søren Fournais, Mathieu Lewin, Jan Philip Solovej. The semi-classical limit of large fermionic systems. Calculus of Variations and Partial Differential Equations, 2018, pp.57-105. ⟨10.1007/s00526-018-1374-2⟩. ⟨hal-01211494v2⟩