  We calculate the thermodynamic entropy of the mean-field $\phi^4$ spin model
in the microcanonical ensemble as a function of the energy and magnetization of
the model. The entropy and its derivative are obtained from the theory of large
deviations, as well as from Rugh's microcanonical formalism, which is
implemented by computing averages of suitable observables in microcanonical
molecular dynamics simulations. Our main finding is that the entropy is a
concave function of the energy for all values of the magnetization, but is
nonconcave as a function of the magnetization for some values of the energy.
This last property implies that the magnetic susceptibility of the model can be
negative when calculated microcanonically for fixed values of the energy and
magnetization. This provides a magnetization analog of negative heat
capacities, which are well-known to be associated in general with the
nonequivalence of the microcanonical and canonical ensembles. Here, the two
ensembles that are nonequivalent are the microcanonical ensemble in which the
energy and magnetization are held fixed and the canonical ensemble in which the
energy and magnetization are fixed only on average by fixing the temperature
and magnetic field.
