Mean‐Field and Classical Limit for the N‐Body Quantum Dynamics with Coulomb Interaction

Abstract

This paper proves the validity of the joint mean-field and classical limit of the quantum N -body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality from [S. Serfaty, with an appendix of M. Duerinckx arXiv:1803.08345v3 [math.AP]].

Mean-Field Limits in Statistical Dynamics

“…Indeed, Theorem IV.5 in [40] assumes that the Wigner transform of the states considered is bounded in L ∞ ([0, T ], L 2 (R 3 × R 3 )). This incompatible with the monokinetic setting in [29], where the Wigner functions considered converge to a Dirac distribution in the momentum variable. Thus Proposition 2.4 in [29] and Theorem IV.5 in [40] both establish the validity of the classical limit of the Hartree equation, but in radically different asymptotic regimes.…”

“…Theorem 3.8. [29] Let ρ in ∈ H 4 (R 3 ) ∩ P(R 3 ) and u in ∈ L ∞ (R 3 ) 3 be such that ∇u in ∈ H 4 (R 3 ) 3 .…”

“…(2) The method of [29] based on Serfaty's inequality can be used to derive rigorously the Vlasov-Poisson system from the Hartree equation in the Coulomb case and in the monokinetic setting: see Proposition 2.4 in [29]. (This is the right vertical arrow in the diagram at the beginning of lecture 3, in other words, the classical limit of the Hartree equation leading to the Vlasov-Poisson system).…”

“…(See also [8] in the case of regular potentials.) There is however a fundamental difference between Proposition 2.4 in [29] and Theorem IV.5 in [40]. Indeed, Theorem IV.5 in [40] assumes that the Wigner transform of the states considered is bounded in L ∞ ([0, T ], L 2 (R 3 × R 3 )).…”

Convergence Towards the Vlasov–Poisson Equation from the N-Fermionic Schrödinger Equation

“…They proved the limit from the Schrödinger equation to Vlasov, in which the interaction potential V was assumed to be analytic in [52] and C 2 in [64]. The rate of convergence of the combined limit in terms of the Wasserstein (pseudo)distance was obtained in [31][32][33]. In fact, the authors studied the rate of convergence in terms of the Wasserstein distance by treating the Vlasov equation as a transport equation and applying the Dobrushin estimate with appropriately chosen initial data.…”