Browsing by Author "Medani, Mohammed"

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    Ultra-cold gases in Low Dimensionality
    (BENAROUS MOHAMED / HOCINE AHMED, 2026) Medani, Mohammed
    This thesis investigates the thermodynamic properties of ideal Bose gases within the framework of the Dunkl formalism, a generalization of quantum mechanics based on the deformed Heisenberg algebra introduced through the Wigner-Dunkl differential-difference operator. Starting from the mathematical foundations of deformed algebras and revisiting the seminal contributions of Wigner, Yang, and Dunkl, we systematically extend the standard Bose-Einstein condensation theory to the Dunkl-deformed setting, covering both homogeneous and confined systems in arbitrary spatial dimension D. For ideal Bose gases confined by general power-law trapping potentials, we show that all thermodynamic quantities depend solely on a single universal parameter s that encoding the combined effects of dimensionality and trap geometry η. This reveals the existence of universality classes applicable to any power-law potential regardless of its specific form. Bose-Einstein condensation occurs exclusively for s > 1 , consistently with the Mermin-Wagner-Hohenberg theorem, and the BEC transition remains second order for all s ̸= 2, while s = 2 it exhibits a continuous transition of Berezinskii-Kosterlitz-Thouless type. The Dunkl deformation parameter ν tunes the thermodynamic behavior continuously, and thermodynamic consistency requires 0 < ν ≤ 2 , a constraint shown to hold for arbitrary regular potentials in any dimension. These results establish a unified description of Dunkl-deformed Bose gases and clarify the fundamental interplay between confinement geometry and algebraic deformation.