  Systems of highly degenerate ordered or frozen state may exhibit inverse
melting (reversible crystallization upon heating) or inverse freezing
(reversible glass transition upon heating). This phenomena is reviewed, and a
list of experimental demonstrations and theoretical models is presented. A
simple spin model for inverse melting is introduced and solved analytically for
infinite range, constant paramagnetic exchange interaction. The random exchange
analogue of this model yields inverse freezing, as implied by the analytic
solution based on the replica trick. The qualitative features of this system
(generalized Blume-Capel spin model) are shown to resemble a large class of
inverse melting phenomena. The appearance of inverse melting is related to an
exact rescaling of one of the interaction parameters that measures the entropy
of the system. For the case of almost degenerate spin states perturbative
expansion is presented, and the first three terms correspond to the empiric
formula for the Flory-Huggins $\chi$ parameter in the theory of polymer melts.
Possible microscopic origin of this $\chi$ parameter and the limitations of the
Flory-Huggins theory where the state degeneracy is associated with the
different conformations of a single polymer or with the spatial structures of
two interacting molecules are discussed.
