Statistique Classique Et Quantique De Systèmes Fractionnaires

Résumé: In this study, our main objective was to analyze the thermodynamic properties of classical and quantum statistical systems described by the fractional Hamiltonian Hα. We started by establishing the necessary theoretical foundations to develop the theory of fractional derivation. Then, we explored fractional quantum mechanics by presenting its fundamental principles and discussing some applications of the fractional Schrodinger equation. We also studied classical and quantum statistical problems within the framework of fractional quantum mechanics. Initially, we introduced the partition function ZN for a gas system composed of N independent fractional quantum oscillators in a D-dimensional space. Using the three definitions of fractional derivatives (Liouville, Riemann-Liouville, and Caputo), we calculated the partition function for each definition and applied it to a three-dimensional quantum oscillator. This application allowed us to demonstrate that the three definitions of fractional derivative generally lead to different partition functions, and therefore different thermodynamic properties. We also focused on solving the fractional Liouville equation using the Riemann- Liouville and Caputo derivatives for systems with non-integer power laws in their Hamiltonians. Based on the fractional Liouville equation, we first used the Riemann- Liouville derivative to calculate the density function (DF) of the classical ideal gas. The results showed that the DF depends on both the momentum p and position q. On the other hand, when we used the Caputo derivative, we found that the DF does not depend on (p, q) and remains constant. Furthermore, we extended this study to a gas system composed of N fractional oscillators in a one-dimensional space, where the DF of the system is influenced by the type of derivative used

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