Abstract
We study the convergence of the Schrödinger equation, when the Planck constant tends to 0. Our analysis leads us to introduce non-discerned particles in classical mechanics and discerned particles in quantum mechanics. These non-discerned particles in classical mechanics correspond to an action and a density which verify the statistical Hamilton-Jacobi equations. The indiscernability of classical particles provides a very simple and natural explanation to the Gibbs paradox. We then consider the case of a large number of identical non-discerned interacting particles modeled by a mean field. In classical mechanics these particles satisfy the mean field Hamilton-Jacobi equations. We show how the analysis of non-discerned particles in classical mechanics can be fruitfully applied to some other fields. In economics, we show that the theory of mean field games, where non-discerned agents are considered interacting with one another, is the analogue of mean field Hamilton-Jacobi equations.
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References
Greiner, W., Neise, L., Stöcker, H.: Thermodynamics and Statistical mechanics. Springer, New York (1995)
Madelung, E.: Zeit. Phys. 40, 322–326 (1926)
Gondran, M., Gondran, A.: Discerned and non-discerned particles in classical mechanics and convergence of quantum mechanics to classical mechanics. Annales de la Fondation Louis de Broglie 36, 117–135 (2011)
Gondran, M., Gondran, A.: The two limits of the Schrödinger equation in the semi-classical approximation: discerned and non-discerned particles in classical mechanics. In: Proceeding of AIP, Conference Foundations of Probability and Physics 6, Växjö, Sweden, vol. 1424 (June 2011)
Schrödinger, E.: Naturwissenschaften 14, 664–666 (1926)
de Broglie, L.: J. de Phys. 8, 225–241 (1927)
Bohm, D.: Phys. Rev. 85, 166–193 (1952)
Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1977)
Lasry, J.-M., Lions, P.L.: Jeux à champ moyen I:le cas stationnaire. C. R. Acad. Sci. Paris 343(9) (2006)
Lasry, J.-M., Lions, P.L.: Jeux à champ moyen II: horizon fini et contrôle optimal. C. R. Acad. Sci. Paris 343(10) (2006)
Lasry, J.-M., Lions, P.L.: Mean Field Games. Japanese Journal of Mathematics 2(1) (March 2007)
Guéant, O.: Mean Fields Games and Applications to Economics - Secondary topic: Discount rates and sustainable development, Ecole doctorale EDDIMO, Centre de Recherche CEREMADE, Université Paris Dauphine (June 30, 2009)
Fessler, D., Lautier, D., Lasry, J.-M.: The economics of sustainable development. Ed. Economica, 366 pages (2010) ISBN 978-2-7178-5851-8
Guéant, O.: A reference case for mean field games models. Cahier de la Chaire Finance et Développement Durable (10) (June 30, 2009)
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Gondran, M., Lepaul, S. (2012). Indiscernability and Mean Field, a Base of Quantum Interaction. In: Busemeyer, J.R., Dubois, F., Lambert-Mogiliansky, A., Melucci, M. (eds) Quantum Interaction. QI 2012. Lecture Notes in Computer Science, vol 7620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35659-9_20
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DOI: https://doi.org/10.1007/978-3-642-35659-9_20
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