Ultra-cold gases in Low Dimensionality
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Date
2026
Authors
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BENAROUS MOHAMED / HOCINE AHMED
Abstract
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.
Description
Doctoral Thesis in Theoretical Physics