  Based on statistical mechanics, a macroscopically homogeneous system, i.e., a
single phase in the present context, is composed of many independent
configurations that the system embraces. The macroscopical properties of the
system are determined by the properties and statistical probabilities of those
configurations with respect to external conditions. The volume of a single
phase is thus the weighted sum of the volumes of all configurations.
Consequently, the derivative of the volume to temperature of a single phase
depends on both the derivatives of the volumes of every configuration to
temperature and the derivatives of their statistical probabilities to
temperature with the latter introducing non-linear emergent behaviors. It is
shown that the derivative of the volume to temperature of the single phase can
be negative, i.e., negative thermal expansion (NTE), due to the
symmetry-breaking non-ground-state configurations with smaller volumes than
that of the ground-state configuration and the rapid increase of the
statistical probabilities of the former, and NTE can be predicted without
fitting parameters from the zentropy theory that combines quantum mechanics and
statistical mechanics with the free energy of each configuration predicted from
quantum mechanics and the partition function of each configuration calculated
from its free energy.
