Nonextensive statistics for a 2D electron gas in noncommutative spaces

TL;DR

The paper addresses the thermodynamics of a two dimensional electron gas in a noncommutative plane under a perpendicular magnetic field and in plane electric field with harmonic confinement by employing Tsallis nonextensive statistics. It adapts the Hilhorst integral transform to derive q generalised partition functions and related thermodynamic quantities from the standard ensemble while accounting for noncommutative geometry. The authors show that the interplay between the nonextensivity parameter and the noncommutativity parameter yields new thermodynamic regimes and anomalous magnetic responses, and they discuss the consequences for the limit q to 1 and for the physical interpretation of the q theta parameter space. They also establish validity domains, convergence criteria and a complementarity principle between q and theta, providing a framework that can be extended to other quantum systems with modified phase spaces. The work has potential implications for quantum Hall like systems and other condensed matter or quantum gravity inspired models where nonextensive statistics and NC geometry play a role.

Abstract

This work investigates a quantum system described by a Hamiltonian operator in a two dimensional noncommutative space. The system consists of an electron subjected to a perpendicular magnetic field $\mathbf{B}$, coupled to a harmonic potential and an external electric field $\mathbf{E}$, within the context of non-extensive statistical thermodynamics. The noncommutative geometry introduces a fundamental minimal length that modifies the phase space structure. The thermodynamics of this quantum system is developed within the framework of Tsallis statistics through the derivation of $q$-generalized versions of the partition function, magnetization, and magnetic susceptibility, following the application of a generalized Hilhorst transformation adapted to non-commutative geometry. The combined effects of the non-extensivity parameter $q$ and the noncommutativity parameter $θ$ are analyzed by considering the limit $q \rightarrow 1$, revealing new thermodynamic regimes and anomalous electromagnetic properties specific to quantum systems in non-commutative geometry.

Figures (6)

• Figure 1: 3D surface plot of the partition function $Z_q(\beta)$ for an electron in a noncommutative plane. The surface shows the variation of $Z_q$ as a function of the non-extensivity parameter $q$ and the inverse temperature $\beta$.
• Figure 2: Contour plots of $Z_q(\beta)$: filled contours (left) and line contours (right). The dashed red line indicates the Boltzmann-Gibbs limit ($q=1$).
• Figure 3: Cross-sectional analysis of $Z_q(\beta)$: (top left) $Z_q$ vs. $\beta$ for different $q$, (top right) $Z_q$ vs. $q$ for different $\beta$, (bottom left) $\log(Z_q)$ vs. $\beta$, (bottom right) $Z_q/Z_{q=1}$ ratio vs. $\beta$.
• Figure 4: Sensitivity of $Z_q(\beta)$ to external parameters: (top left) $\theta$ dependence, (top right) $E$ dependence, (bottom left) $\omega_c$ dependence, (bottom right) $\theta$-$\omega_c$ joint sensitivity. Reference values: red dashed lines/black markers.
• Figure 5: Dependence of normalized magnetization $M_{q}^{(\theta)}$ on the deformation parameter $q$ for different orientation angles $\theta$ in a two-dimensional $q$-deformed spin system.