On the Magnetization of a Charged Bose Gas in the Canonical Ensemble

Consider a charged Bose gas without self-interactions, confined in a three dimensional cubic box of side L ≥ 1 and subjected to a constant magnetic field B 6= 0. If the bulk density of particles ρ and the temperature T are fixed, then define the canonical magnetization as the partial derivative with respect to B of the reduced free energy. Our main result is that it admits thermodynamic limit for all strictly positive ρ, T and B. It is also proven that the canonical and grand canonical magnetizations (the last one at fixed average density) are equal up to the surface order corrections.