Indiscernability and Mean Field, a Base of Quantum Interaction

  • Michel Gondran
  • Sébastien Lepaul
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7620)


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.


indiscernability Gibbs paradox mean field mean field game Hamilton-Jacobi equations 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michel Gondran
    • 1
    • 2
  • Sébastien Lepaul
    • 1
    • 2
  1. 1.University Paris DauphineParisFrance
  2. 2.EDF R et DClamart CedexFrance

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