  The thermodynamical analogy of density functional theory, which is an organic
part of the spin-independent version of the theory, is reconsidered for its
spin-polarized generalization in view of the recently uncovered nonuniqueness
of the external magnetic field B(r) corresponding to a given pair of density
n(r) and spin density n_s(r). For ground states, the nonuniqueness of B(r)
implies the nondifferentiability of the energy functional E[n,n_s] with respect
to n_s(r). It is shown, on the other hand, that this nonuniqueness allows the
existence of the one-sided derivatives of E[n,n_s] with respect to n_s(r).
Although the N-electron ground state can always be obtained from the
minimization of E[n,n_s] without any constraint on the spin number N_s, the
Lagrange multiplier mu_s associated with the fixation of N_s does not vanish
even for ground states. Rather, mu_s is identified as the left- or right-side
derivative of the total energy with respect to N_s. This justifies the
interpretation of mu_s as a (spin) chemical potential, which is the cornerstone
of the thermodynamical analogy.
