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arXiv:math-ph/0605019

Generalized susceptibilities for a perfect quantum gas

Philippe Briet, Horia D. Cornean, Delphine Louis

Published May 5, 2006• 6 citations
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Abstract

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 $Λ$ with side $L\geq 1$, and subjected to a constant magnetic field of intensity $ B \geq 0 $. Define the grand canonical generalized susceptibilities $χ_L^N$, $N\geq 1$, as successive partial derivatives with respect to $B$ of the grand canonical pressure $P_L$. Denote by $P_{\infty}$ the thermodynamic limit of $P_L$. Our main result is that $χ_L^N$ admit as thermodynamic limit the corresponding partial derivatives with respect to $B$ of $P_{\infty}$. In this paper we only give the main steps of the proofs, technical details will be given elsewhere.

DOI
10.48550/arxiv.math-ph/0605019

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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≥0B \geq 0B≥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.
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).
We investigate the isobar of an ideal Bose gas confined in a cubic box within the grand canonical ensemble for a large yet finite number of particles, NNN. After solving the equation of the spinodal curve, we derive precise formulas for the supercooling and the superheating temperatures that reveal an ${N}^{\ensuremath{-}1/3}$ or ${N}^{\ensuremath{-}1/4}$ power correction to the known Bose-Einstein condensation temperature in the thermodynamic limit. Numerical computations confirm the accuracy of our analytical approximation, and further show that the isobar zigzags on the temperature-volume plane if $N\ensuremath{\ge}14\phantom{\rule{0.16em}{0ex}}393$. In particular, for the Avogadro's number of particles, the volume expands discretely about 105{10}^{5}105 times. Our results quantitatively agree with a previous study on the canonical ensemble within 0.10.1%0.1 error.
Many-particle systems pose commonly known computational challenges in quantum theory. The obstacles arise from the difficulty in finding sets of eigenvalues and eigenvectors of the underlying Hamiltonian while enforcing fermion or boson statistics, not to mention the prohibitive increase in the computational cost with the system’s size. The first obvious step in this direction is to elaborate the theory for Fermi or Bose gases without inter-particle interactions. The traditional approach to the work is with the ideal gases confined in a cubic container with impenetrable walls (in arbitrary dimensions). This approach allows one to find the particle’s spectra and compute all thermodynamic quantities of the confined gas. In the present work, we consider the gas confined in a spherical container (in other words, an infinitely deep spherical potential well in D dimensions), solving the corresponding Schrödinger equation using zero boundary conditions. We address the case of a finite number of particles N, either bosons or fermions, in the spherical potential box, as well as the thermodynamic limit, corresponding to N → ∞. Owing to Weyl’s relations, in the latter limit, the results do not depend on the shape of the box and thus approach the commonly known ones valid in the infinite space. Owing to the underlying SO(D) symmetry, we are dealing with particles carrying a well-defined angular momentum that, together with sorted energy eigenvalues, imparts a shell structure to the system.