  The Li$_2$S-P$_2$S$_5$ pseudo-binary system has been a valuable source of
promising superionic conductors, with $\alpha$-Li$_3$PS$_4$,
$\beta$-Li$_3$PS$_4$, HT-Li$_7$PS$_6$, and Li$_7$P$_3$S$_{11}$ having excellent
room temperature Li-ion conductivity > 0.1 mS/cm. The metastability of these
phases at ambient temperature motivates a study to quantify thermodynamic
accessibility. Through calculating the electronic, configurational, and
vibrational sources of free energy from first principles, a phase diagram of
the crystalline Li$_2$S-P$_2$S$_5$ space is constructed. Well-established phase
stability trends from experiments are recovered, such as polymorphic phase
transitions in Li$_7$PS$_6$ and Li$_3$PS$_4$, and the metastability of
Li$_7$P$_3$S$_{11}$ at high temperature. At ambient temperature, it is
predicted that all superionic conductors in this space are indeed metastable,
but thermodynamically accessible. Vibrational and configurational sources of
entropy are shown to be essential towards describing the stability of
superionic conductors. New details of the Li sublattices are revealed, and are
found to be crucial towards accurately predicting configurational entropy. All
superionic conductors contain significant configurational entropy, which
suggests an inherent correlation between superionic conductivity and high
configurational entropy.
