Abstract
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 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, 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.
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References
Adami R., Golse F., Teta A.: Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys. 127, 1193–1220 (2007)
Ammari, Z., Falconi, M., Pawilowski, B.: On the rate of convergence for the mean field approximation of many-body quantum dynamics. Commun. Math. Sci. arXiv:1411.6284 (preprint)
Bardos C., Erdös L., Golse F., Mauser N., Yau H.-T.: Derivation of the Schrödinger–Poisson equation from the quantum N-body problem. C. R. Math. Acad. Sci. Paris 334, 515–520 (2002)
Bardos C., Golse F., Mauser N.: Weak coupling limit of the N-particle Schrödinger equation. Method. Appl. Anal. 7, 275–293 (2000)
Benedikter N., Porta M., Schlein B.: Mean-field evolution of Fermionic systems. Commun. Math. Phys. 331, 1087–1131 (2014)
Braun W., Hepp K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56, 101–113 (1977)
Dobrushin R.: Vlasov equations. Funct. Anal. Appl. 13, 115–123 (1979)
Erdös L., Yau H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5, 1169–1205 (2001)
Erdös L., Schlein B., Yau H.-T.: Derivation of the cubic non-linear Schrdinger equation from quantum dynamics of many-body systems. Invent. Math. 167, 515–614 (2007)
Erdös L., Schlein B., Yau H.-T.: Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate. Ann. Math. (2) 172, 291–370 (2010)
Fournier N., Guillin A.: On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Rel. Fields 162(3), 707–738 (2015)
Fröhlich J., Knowles A., Schwarz S.: On the mean-field limit of Bosons with Coulomb two-body interaction. Commun. Math. Phys. 288, 1023–1059 (2009)
Gérard P., Markowich P., Mauser N., Poupaud F.: Homogenization limit and Wigner transforms. Commun. Pure Appl. Math. 50, 323–379 (1997)
Graffi S., Martinez A., Pulvirenti M.: Mean-field approximation of quantum systems and classical limit. Math. Models Methods Appl. Sci. 13, 59–73 (2003)
Golse F., Mouhot C., Ricci V.: Empirical measures and Vlasov hierarchies. Kin. Rel. Models 6, 919–943 (2013)
Hauray M., Jabin P.-E.: Particle approximation of Vlasov equations with singular forces. Ann. Sci. Ecol. Norm. Sup. 48, 891–940 (2015)
Knowles A., Pickl P.: Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298, 101–138 (2010)
Lazarovici, D.: The Vlasov–Poisson dynamics as the mean-field limit of rigid charges. arXiv:1502.07047 (preprint)
Lazarovici, D., Pickl, P.: A mean-field limit for the Vlasov–Poisson system. arXiv:1502.04608 (preprint)
Lions P.-L., Paul T.: Sur les mesures de Wigner. Rev. Math. Iber. 9, 553–618 (1993)
Loeper G.: Uniqueness of the solution to the Vlasov–Poisson system with bounded density. J. Math. Pure. Appl. 86, 68–79 (2006)
Mischler S., Mouhot C., Wennberg B.: A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. Probab. Theory Rel. Fields 161, 1–59 (2015)
Narnhofer H., Sewell G.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79, 9–24 (1981)
Neunzert, H., Wick, J.: Die Approximation der Lösung von Integro–Differentialgleichungen durch endliche Punktmengen; Lecture Notes in Math. vol. 395, pp. 275–290, Springer, Berlin (1974)
Pezzotti F., Pulvirenti M.: Mean-field limit and semiclassical expansion of a quantum particle system. Ann. Henri Poincaré 10, 145–187 (2009)
Pickl P.: A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97, 151–164 (2011)
Pickl P.: Derivation of the time dependent Gross–Pitaevskii equation without positivity condition on the interaction. J. Stat. Phys. 140, 76–89 (2010)
Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291, 31–61 (2009)
Spohn H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52, 600–640 (1980)
Spohn H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3, 445–455 (1981)
Villani C.: Topics in Optimal Transportation. American Mathematical Soc, Providence (RI) (2003)
Villani C.: Optimal Transport: Old and New. Springer, Berlin (2009)
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Communicated by H. Spohn
In memory of Louis Boutet de Monvel (1941–2014)
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Golse, F., Mouhot, C. & Paul, T. On the Mean Field and Classical Limits of Quantum Mechanics. Commun. Math. Phys. 343, 165–205 (2016). https://doi.org/10.1007/s00220-015-2485-7
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DOI: https://doi.org/10.1007/s00220-015-2485-7