On the Mean Field and Classical Limits of Quantum Mechanics

By François Golse, Clément Mouhot, Thierry Paul. Communications in Mathematical Physics, 2015

159 citations recorded by the source corpus.

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 equation. 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 C1,1 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, 115–123, (1979)]. Our approach to 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.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/875861147a1a7c3e7de821a103dc3a37bd44f8d0

Citations in the corpus

• Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, unknown): “Contrary to usual laws of large numbers, that have been used for Mean Field limits recently in [26] with K ∈W 1,∞ loc , RN here exhibits a double sum so that a priori RN = O(N) and the challenge in Theorem 2 is to prove that in fact RN = O(1).”