  It has been observed in both natural and man-made materials that volume
sometimes decreases with increasing temperature. Though mechanistic
understanding has been gained for some individual materials, a general answer
to the question "Why does volume sometimes decrease with the increase of
temperature?" remains lacking. Based on the thermodynamic relation that the
derivative of volume with respect to temperature, i.e., thermal expansion, is
equal to the negative derivative of entropy with respect to pressure, we
developed a general theory in terms of multiscale entropy to understand and
predict the change of volume as a function of temperature, which is termed as
zentropy theory in the present work. It is shown that a phase at high
temperatures is a statistical representation of the ground-state stable and
multiple nonground-state metastable configurations. It is demonstrated that
when the volumes of the major nonground-state configurations are smaller than
that of the ground-state configuration, the volume of the phase may decrease
with the increase of temperature in certain ranges of temperature-pressure
combinations, depicting the negative divergency of thermal expansion at the
critical point. As examples, positive and negative divergencies of thermal
expansion are predicted at the critical points of Ce and Fe3Pt, respectively,
along with the temperature and pressure ranges for abnormally positive and
negative thermal expansion. The authors believe that the zentropy theory is
applicable to predict anomalies of other physical properties of phases because
the change of entropy drives the responses of a system to external stimuli.
