The mean-field limit for the dynamics of large particle systems
Golse, François
Journées équations aux dérivées partielles, (2003), p. 1-47 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

This short course explains how the usual mean-field evolution PDEs in Statistical Physics - such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations - are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.

@article{JEDP_2003____A9_0,
     author = {Golse, Fran\c cois},
     title = {The mean-field limit for the dynamics of large particle systems},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2003},
     pages = {1-47},
     doi = {10.5802/jedp.623},
     mrnumber = {2050595},
     zbl = {02079444},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2003____A9_0}
}

Golse, François. The mean-field limit for the dynamics of large particle systems. Journées équations aux dérivées partielles,  (2003), pp. 1-47. doi : 10.5802/jedp.623. http://gdmltest.u-ga.fr/item/JEDP_2003____A9_0/

[25] R. Adami, C. Bardos, F. Golse, A. Teta, Towards a rigorous derivation of the cubic nonlinear Schrödinger equation in dimension 1, preprint.

[26] C. Bardos, L. Erdös, F. Golse, N. Mauser, H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum N-body problem, C. R. Acad. Sci. Sér. I Math 334 (2002), 515-520. | MR 1890644 | Zbl 1018.81009

[27] C. Bardos, F. Golse, A. Gottlieb, N. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, to appear in J. de Math. Pures et Appl. 82 (2003). | MR 1996777 | Zbl 1029.82022

[28] C. Bardos, F. Golse, A. Gottlieb, N. Mauser, Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Potential, in preparation.

[30] C. Bardos, F. Golse, N. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), no. 2, 275-293. | MR 1869286 | Zbl 1003.81027

[31] A. Bove, G. Daprato, G. Fano, An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction, Commun. Math. Phys. 37 (1974), 183-191. | MR 424069 | Zbl 0303.34046

[32] A. Bove, G. Daprato, G. Fano, On the Hartree-Fock time-dependent problem, Comm. Math. Phys. 49 (1976), 25-33. | MR 456066

[33] W. Braun, K. Hepp, The Vlasov Dynamics and Its Fluctuations in the 1/N Limit of Interacting Classical Particles; Commun. Math. Phys. 56 (1977), 101-113. | MR 475547 | Zbl 1155.81383

[34] E. Caglioti, P.-L. Lions, C. Marchioro, M. Pulvirenti: A special class of flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys. 143 (1992), 501-525. | MR 1145596 | Zbl 0745.76001

[35] E. Cancès, C. Le Bris, On the time-dependent Hartree-Fock equations coupled with a classical nuclear dynamics, Math. Models Methods Appl. Sci. 9 (1999), 963-990. | MR 1710271 | Zbl 1011.81087

[36] I. Catto, C. Le Bris, P.-L. Lions, The mathematical theory of thermodynamic limits: Thomas-Fermi type models, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, (1998). | MR 1673212 | Zbl 0938.81001

[37] J.M. Chadam, R.T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Mathematical Phys. 16 (1975), 1122-1130. | MR 413843 | Zbl 0299.35084

[38] L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation; preprint. | MR 1787105 | Zbl 1162.82316

[39] R. Dobrushin, Vlasov equations; Funct. Anal. Appl. 13 (1979), 115-123. | MR 541637 | Zbl 0422.35068

[40] L. Erdös, H.-T. Yau: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), 1169-1205. | MR 1926667 | Zbl 1014.81063

[41] G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas, Nota int. no. 358, Istituto di Fisica, Università di Roma, (1972). Reprinted in Statistical Mechanics: a Short Treatise, pp. 48-55, Springer-Verlag Berlin-Heidelberg (1999)

[42] M. Kiessling, Statistical mechanics of classical particles with logarithmic interactions; Commun. Pure Appl. Math. 46 (1993), 27-56. | MR 1193342 | Zbl 0811.76002

[43] F. King, PhD Thesis, U. of California, Berkeley 1975.

[44] L. Landau, E. Lifshitz: Mécanique quantique; Editions Mir, Moscou 1967.

[45] L. Landau, E. Lifshitz: Théorie quantique relativiste, première partie; Editions Mir, Moscou 1972. | MR 389019

[46] L. Landau, E. Lifshitz: Physique statistique, deuxième partie; Editions Mir, Moscou 1990.

[47] O. Lanford: Time evolution of large classical systems, in ``Dynamical systems, theory and applications" (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 1-111. Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975. | MR 479206 | Zbl 0329.70011

[48] C. Marchioro, M. Pulvirenti Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag (1994). | MR 1245492 | Zbl 0789.76002

[50] V. Maslov: Equations of the self-consistent field; J. Soviet Math. 11 (1979), 123-195. | MR 489604 | Zbl 0408.45010

[51] H. Narnhofer, G.L. Sewell, Vlasov hydrodynamics of a quantum mechanical model, Comm. Math. Phys. 79 (1981), 9-24. | MR 609224

[52] H. Neunzert The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles; Trans. Fluid Dynamics 18 (1977), 663-678.

[52] L. Nirenberg An abstract form of the nonlinear Cauchy-Kowalewski theorem; J. Differential Geometry 6 (1972), 561-576. | MR 322321 | Zbl 0257.35001

[53] L. Onsager Statistical hydrodynamics, Supplemento al Nuovo Cimento 6 (1949), 279-287. | MR 36116

[54] T. Nishida A note on a theorem of Nirenberg; J. Differential Geometry 12 (1977), 629-633. | MR 512931 | Zbl 0368.35007

[55] H. Spohn, Kinetic Equations from Hamiltonian Dynamics: Markovian Limits, Rev. Modern Phys. 52 (1980), 569-615. | MR 578142 | Zbl 0399.60082