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The computation of the energy of a diluted Fermi or Bose gas has been well-known by physicists since the 50's-60's. During the last twenty years, many mathematical results have established a rigourous basis for these formulas. In my talk, I will first present the physical context, especially the link between classical and quantum mechanics, before explaining the rigorous formulation of the many-body quantum mechanics. Finally, I will try to present the model I am studying.