  To address the effects of lattice anharmonicity across the perovskite to
post-perovskite transition in MgSiO$_3$, we conduct calculations using the
phonon quasiparticle (PHQ) approach. The PHQ is based on \textit{ab initio}
molecular dynamics and, in principle, captures full anharmonicity. Free
energies in the thermodynamic limit ($N \rightarrow \infty$) are computed using
temperature-dependent quasiparticle dispersions within the phonon gas model.
Systematic results on anharmonic thermodynamic properties and phase boundary
are reported. Both the local density approximation (LDA) and the generalized
gradient approximation (GGA) calculations are performed to provide confident
constraints on these properties. Anharmonic effects are demonstrated by
comparing results with those obtained using the quasiharmonic approximation
(QHA). The inadequacy of the QHA is indicated by its overestimation of thermal
expansivity and thermodynamic Gr\"{u}neisen parameter and its converged
isochoric heat capacity in the high-temperature limit. The PHQ phase boundary
has a Clapeyron slope ($dP/dT$) that increases with temperature. This result
contrasts with the nearly zero curvature of the QHA phase boundary.
Anharmonicity bends the phase boundary to lower temperatures at high pressures.
Implications for the double-crossing of the phase boundary by the mantle
geotherm are discussed.
