Magnetic systems in the generalized canonical ensemble

The problem of spin-1/2 non-interacting particles in a constant magnetic field is revisited. Despite being a workhorse model for introducing magnetism in statistical mechanics, there are several aspects of this problem that can be confusing to students: the Hamiltonian of the system (H = −µ B0 or H = 0?), the expression of the magnetic internal energy variation (dU = T dS + B0 · dm or dU = T dS − m · dB0 ?), the statistical ensemble to analyze the system and the corresponding thermodynamic potential derived from it (a Helmholtz free energy or a Gibbs potential?). Ultimately, all these questions are related to a very essential point: the need to exactly establish what the system under study and the boundaries are. Using a generalized Gibbs canonical ensemble a rigorous approach to this problem addressing all those questions is offered. To deepen the discussion, the analysis is extended to analogous systems, including classical electric dipoles in a uniform electric field and a polymer chain under constant tension.