On the Mean Field and Classical Limits of Quantum Mechanics

François Golse (1) , Clément Mouhot (2) , Thierry Paul (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics

The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schrödinger equation and of the mean field Hartree equations. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.

Mots clés: Hartree equation; Liouville equation; Vlasov equation; Mean field limit; Classical limit; Monge-Kantorovich distance; Schrödinger equation

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

François Golse, Clément Mouhot, Thierry Paul. On the Mean Field and Classical Limits of Quantum Mechanics. 2015. ⟨hal-01119132v2⟩