  The phase diagram of numerous materials of technological importance features
high-symmetry high-temperature phases that exhibit phonon instabilities.
Leading examples include shape-memory alloys, as well as ferroelectric,
refractory, and structural materials. The thermodynamics of these phases have
proven challenging to handle by atomistic computational thermodynamic
techniques, due to the occurrence of constant anharmonicity-driven hopping
between local low-symmetry distortions, while maintaining a high-symmetry
time-averaged structure. To compute the free energy in such phases, we propose
to explore the system's potential-energy surface by discrete sampling of local
minima by means of a lattice gas Monte Carlo approach and by continuous
sampling by means of a lattice dynamics approach in the vicinity of each local
minimum. Given the proximity of the local minima, it is necessary to carefully
partition phase space by using a Voronoi tessellation to constrain the domain
of integration of the partition function, in order to avoid double-counting
artifacts and enable an accurate harmonic treatment near each local minima. We
consider the bcc phase of titanium as a prototypical examples to illustrate our
approach.
