  The vibrational entropy of a solid at finite temperature is investigated from
the perspective of information theory. Ab initio molecular dynamics (AIMD)
simulations generate ensembles of atomic configurations at finite temperature
from which we obtain the $N$-body distribution of atomic displacements,
$\rho_N$. We calculate the information-theoretic entropy from the expectation
value of $\ln{\rho_N}$. At a first level of approximation, treating individual
atomic displacements independently, our method may be applied using
Debye-Waller B-factors, allowing diffraction experiments to obtain an upper
bound on the thermodynamic entropy. At the next level of approximation we
correct the overestimation through inclusion of displacement covariances. We
apply this approach to elemental body-centered cubic sodium and face-centered
cubic aluminum, showing good agreement with experimental values above the Debye
temperatures of the metals. Below the Debye temperatures we extract an
effective vibrational density of states from eigenvalues of the covariance
matrix, and then evaluate the entropy quantum mechanically, again yielding good
agreement with experiment down to low temperatures. Our method readily
generalizes to complex solids, as we demonstrate for a high entropy alloy.
Further, our method applies in cases where the quasiharmonic approximation
fails, as we demonstrate by calculating the HCP/BCC transition in Ti.
