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arXiv:1005.1584 · Journal of Physics A Mathematical and Theoretical

Diamagnetism of quantum gases with singular potentials

Philippe Briet, Horia D. Cornean, Baptiste Savoie

Published Nov 9, 2010• 6 citations
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Abstract

We consider a gas of quasi-free quantum particles confined to a finite box, subjected to singular magnetic and electric fields. We prove in great generality that the finite volume grand-canonical pressure is analytic with respect to the chemical potential and the intensity of the external magnetic field. We also discuss the thermodynamic limit.

DOI
10.1088/1751-8113/43/47/474008

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We consider a gas of quasi-free quantum particles confined to a finite box, subjected to singular magnetic and electric fields. We prove in great generality that the finite volume grand-canonical pressure is analytic with respect to the chemical potential and the intensity of the external magnetic field. We also discuss the thermodynamic limit.
In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity B. We investigate the Gibbs semigroup associated with the one particle operator at finite volume, and study its Taylor series with respect to the field parameter ω≔eB∕c in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to ω (the so-called generalized susceptibilities).
The system we consider here is a charged fermions gas in the effective mass approximation, and in grand-canonical conditions. We assume that the particles are confined in a three dimensional cubic box ΛΛΛ with side L≥1L\geq 1L≥1, and subjected to a constant magnetic field of intensity $ B \geq 0 $. Define the grand canonical generalized susceptibilities χLNχ_L^NχLN​, N≥1N\geq 1N≥1, as successive partial derivatives with respect to BBB of the grand canonical pressure PLP_LPL​. Denote by P∞P_{\infty}P∞​ the thermodynamic limit of PLP_LPL​. Our main result is that χLNχ_L^NχLN​ admit as thermodynamic limit the corresponding partial derivatives with respect to BBB of P∞P_{\infty}P∞​. In this paper we only give the main steps of the proofs, technical details will be given elsewhere.
We investigate the thermodynamic behavior of a confined ideal quantum gas when subjected to an external magnetic filed ω\omegaω. Roughly speaking, we claim that the derivatives of any order with respect to ω\omegaω of the grand-canonical (canonical) potential admit a certain type of thermodynamic limit. Complete proofs will be given elsewhere.
A charged Bose gas under confinements ensures a phase transition, in contrast with the homogeneous system in an external magnetic field. The present investigation concerns charged Bose gases trapped in a two-dimensional potential as well as neutral Bose gases rotating around the z axis. It is found that the systems exhibit quite different magnetic properties. A diamagnetization is determined in the charged system while the rotating system shows up an uncommon magnetic response. However, both models possess finite magnetic susceptibilities at vanishing field or at stopping rotation.
In this work, we study the thermodynamic functions of quantum gases confined to spaces of various shapes, namely, a sphere, a cylinder and an ellipsoid. We start with the simplest situation, namely, a spinless gas treated within the canonical ensemble framework. As a next step, we consider noninteracting gases (fermions and bosons) with the usage of the grand canonical ensemble description. For this case, the calculations are performed numerically. We also observe that our results may possibly be applied to Bose–Einstein condensate and to helium dimer. Moreover, the bosonic sector, independently of the geometry, acquires entropy and internal energy greater than for the fermionic case. Finally, we also devise a model allowing us to perform analytically the calculations in the case of interacting quantum gases, and, afterwards, we apply it to a cubical box.