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Magnetic response in ideal quantum gases: the thermodynamic limit

Horia D. Cornean

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

We investigate the thermodynamic behavior of a confined ideal quantum gas when subjected to an external magnetic filed $\omega$. Roughly speaking, we claim that the derivatives of any order with respect to $\omega$ of the grand-canonical (canonical) potential admit a certain type of thermodynamic limit. Complete proofs will be given elsewhere.

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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.
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.
Previously developed approach [1 is extended to systems of noninteracting quantum particles in an external magnetic field. Feynman type series for grand potential, number of particles, internal energy and magnetization are derived. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in presence of the homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single particle states. Particular attention is paid to low temperature behavior of fermionic systems.
We investigate the magnetic properties of the ideal Bose gas in both the relativistic and nonrelativistic regimes, and demonstrate its remarkable behavior as a superconductor, where it expels totally an external homogeneous field, and also a new phenomenon, which we call superdiamagnetism, where the gas very nearly does so. These two phenomena are shown to exist in two distinct temperature regimes for both the nonrelativistic and relativistic gases, and are revealed by a careful consideration of the analytic structure of the thermodynamic potential of the system, accessible by the powerful method of the Mellin integral transform. We also comment on other approaches to this problem which have recently appeared.