Etude De Systèmes Statistiques Dans Diverses Dimensions D’espace Basée Sur Une Mécanique Quantique Fractionnaire

Résumé: nthis work, we focus our attention on the study of thermodynamic properties of certain classical and quantum statistical systems that are described by the fractional Hamiltonian H a . In our thesis, firstly, we presented the fundamental tools of the construction of the theory of fractional derivative. Then we give an overview of the fractional quantum mechanics by introducing its principles and some applications of fractional Schrödinger equation then we studied some classical and quantum statistical problems in the context of the fractional quantum mechanics in D dimensions. The partition function Z N and thermodynamic functions for a free ideal gas is calculated. We also calculated the thermodynamic func- tions for the case of a gas of N fractional oscillators heated to a tem- perature T always within the framework of the fractional quantum mechanics using the microcanonical ensemble and the canonical en- semble. In this work, we have also focused on the study of a free and trapped quantum gas of fermions and bosons without interactions (ideal gas) in the fractional case, and we addressed the phase transition problem for the bosons by discussing all thermodynamic quantities including Bose temperature as function of fractionalité of the derivative in the Hamiltonian and we showed that the trapped quantum gases is equi- valent to the free quantum gas. We found that the partition of N fractional quantum oscillators system function is modified and the thermodynamic properties of this system are also modified

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