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

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 $-\infty$ to $\infty$ 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 thermodynamiclimit 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.Comment: 50 pages. Revised version. Accepted for publication in R.M.

A rigorous proof of the Landau-Peierls formula and much more

We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of a non-interacting Bloch electron gas, at fixed temperature and density, for both metals and semiconductors/insulators. In particular, we obtain the Landau-Peierls formula in the low temperature and density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.Comment: 30 pages - Accepted for publication in A.H.

Correlation of clusters: Partially truncated correlation functions and their decay

In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain forests graphs, the connected components of which are tree graphs. We generalize the method introduced by R.A. Minlos and S.K. Poghosyan (1977) in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems.Comment: 31 pages, 2 figures. 2nd revision. Misprints corrected and 1 figure adde

The ideal Bose gas in open harmonic trap systems and its condensation revisited

We rigorously revisit a textbook model used to figure out the Bose-Einstein condensation (BEC) phenomenon created by dilute cold alkali atoms gases in a magnetic-optical trap. It consists of a d-dimensional (d = 1, 2, 3) ideal non-relativistic spin-0 Bose gas confined in a box and trapped in an isotropic harmonic potential. Throughout we review and clarify a series of methods involved in the derivation of the thermodynamics in the grand-canonical situation. To make the derivation consistent with the usual rules of the statistical mechanics, we assign through our open-trap limit approach the role of canonical parameter to a rescaled number of particles (instead of an effective density involving the pulsation of the trap). Within this approach, we formulate an Einstein-like and Penrose-Onsager-like criterion of BEC and show their equivalence. Afterwards, we focus on the spatial localization of the condensate/thermal gas. When dealing with the reduced density matrix, our method is similar to the loop path approach

Gaussian decay for a difference of traces of the Schrödinger semigroup associated to the isotropic harmonic oscillator

This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schrödinger semigroup associated to the quantum isotropic harmonic oscillator. Denoting by H_∞,κ the self-adjoint realization in L 2 (R d ), d ∈ {1, 2, 3} of the Schrödinger operator −(1/2)∆ + (1/2)κ^2*|x|^2, κ > 0 and by H_L,κ, L > 0 the Dirichlet realization in L^2(Λ^d_L) where Λ^d_L := {x ∈ R^d : −L/2 0 has a Gaussian decay in L for L sufficiently large. L The estimate we derive is sharp in the sense that its behavior when κ ↓ 0 and t ↓ 0 is similar to the one given by Tr_(L^2(R^d))*e^(−tH_∞,κ) = (2sinh((κ/2)*t))^(−d). Further, we give a simple application within the framework of quantum statistical mechanics

A rigorous approach to the magnetic response in disordered systems

This paper is a part of an ongoing study on the diamagnetic behavior of a 3-dimensional quantum gas of non-interacting charged particles subjected to an external uniform magnetic field together with a random electric potential. We prove the existence of an almost-sure non-random thermodynamic limit for the grand-canonical pressure, magnetization and zero- field orbital magnetic susceptibility. We also give an explicit formulation of these thermodynamic limits. Our results cover a wide class of physically relevant random potentials which model not only crystalline disordered solids, but also amorphous solids.Comment: 35 pages. Revised version. Accepted for publication in RM