Geodesic
The mean-field limit for the dynamics of large particle systems
Journées équations aux dérivées partielles (2003), article no. 9, 47 p.

Voir la notice de l'acte provenant de la source Numdam

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.

@incollection{JEDP_2003____A9_0,
     author = {Golse, Fran\c{c}ois},
     title = {The mean-field limit for the dynamics of large particle systems},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {9},
     pages = {1--47},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.623},
     mrnumber = {2050595},
     zbl = {02079444},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/jedp.623/}
}
TY  - JOUR
AU  - Golse, François
TI  - The mean-field limit for the dynamics of large particle systems
JO  - Journées équations aux dérivées partielles
PY  - 2003
SP  - 1
EP  - 47
PB  - Université de Nantes
UR  - http://geodesic.mathdoc.fr/articles/10.5802/jedp.623/
DO  - 10.5802/jedp.623
LA  - en
ID  - JEDP_2003____A9_0
ER  - 
%0 Journal Article
%A Golse, François
%T The mean-field limit for the dynamics of large particle systems
%J Journées équations aux dérivées partielles
%D 2003
%P 1-47
%I Université de Nantes
%U http://geodesic.mathdoc.fr/articles/10.5802/jedp.623/
%R 10.5802/jedp.623
%G en
%F JEDP_2003____A9_0
Golse, François. The mean-field limit for the dynamics of large particle systems. Journées équations aux dérivées partielles (2003), article  no. 9, 47 p. doi : 10.5802/jedp.623. http://geodesic.mathdoc.fr/articles/10.5802/jedp.623/

[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. | Zbl | MR

[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). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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). | Zbl | MR

[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. | Zbl | MR

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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

[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. | Zbl | MR

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

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

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

[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. | Zbl | MR

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

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

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

Cité par Sources :