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 greater than or equal to 1 and subjected to a constant magnetic field B not equal 0. If the bulk density of particles p 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 p, 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.