  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.
