Inverse melting is the situation in which a liquid freezes when it is heated isobarically. Both helium isotopes exhibit intervals of inverse melting at low temperature, and published data suggests that isotactic poly (4-methylpentene-1) also displays this unusual phase behavior. Here we propose a statistical mechanical model for inverse melting. It is a decorated modification of the Gaussian core model, in which particles possess a spectrum of thermally activated internal states. Excitation leads to a change in a particle's Gaussian interaction parameters, and this can result in a spatially periodic crystal possessing a higher entropy than the fluid with which it coexists. Numerical solution of the model, using integral equations and the hypernetted chain closure for the fluid phase, and the Einstein model for the solid phases, identifies two types of inverse melting. One mimics the behavior of the helium isotopes, for which the higher-entropy crystal is denser than the liquid. The other corresponds to inverse melting in poly(4-methylpentene-1), where the high-entropy crystal is less dense than the liquid with which it coexists. (C) 2003 American Institute of Physics.