A rigorous proof of the Bohr–van Leeuwen theorem in the semiclassical limit

Savoie, Baptiste (2015) A rigorous proof of the Bohr–van Leeuwen theorem in the semiclassical limit. Reviews in Mathematical Physics, 27 (08). p. 1550019. ISSN 0129-055X

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Abstract

The original formulation of the Bohr-van Leeuwen (BvL) theorem states that, in a uniform magnetic field and in thermal equilibrium, the magnetization of an electron gas in the classical Drude-Lorentz model vanishes identically. This stems from classical statistics which assign the canonical momenta all values ranging from −∞ to ∞ what makes the free energy density magnetic-field-independent. When considering a classical (Maxwell-Boltzmann) interacting electron gas, it is usually admitted that the BvL theorem holds upon condition that the potentials modeling the interactions are particle-velocities-independent and do not cause the system to rotate after turning on the magnetic field. From a rigorous viewpoint, when treating large macroscopic systems one expects the BvL theorem to hold provided the thermodynamic limit of the free energy density exists (and the equivalence of ensemble holds). This requires suitable assumptions on the many-body interactions potential and on the possible external potentials to prevent the system from collapsing or flying apart. Starting from quantum statistical mechanics, the purpose of this article is to give, within the linear-response theory, a proof of the BvL theorem in the semiclassical limit when considering a dilute electron gas in the canonical conditions subjected to a class of translational invariant external potentials.

Item Type:	Article
Uncontrolled Keywords:	Classical magnetism, Diamagnetism, Bohr-van Leeuwen Theorem, Maxwell-Boltzmann statistics, Thermodynamic limit, Ensemble equivalence, Thermodynamic stability, Semiclassical limit, Geometric perturbation theory, Gauge invariant magnetic perturbation theory
Divisions:	School of Theoretical Physics > Preprints
Date Deposited:	05 Oct 2017 19:31
Last Modified:	17 Dec 2022 10:08
URI:	https://dair.dias.ie/id/eprint/322

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