  We have investigated Mg intercalation into orthorhombic V$_2$O$_5$, one of
only three cathodes known to reversibly intercalate Mg ions. By calculating the
ground state Mg$_x$V$_2$O$_5$ configurations and by developing a cluster
expansion for the configurational disorder in $\delta$-V$_2$O$_5$, a full
temperature-composition phase diagram is derived. Our calculations indicate an
equilibrium phase separating behavior between fully demagnesiated
$\alpha$-V$_2$O$_5$ and fully magnesiated $\delta$-V$_2$O$_5$, but also
motivate the existence of potentially metastable solid solution transformation
paths in both phases. We find significantly better mobility for Mg in the
$\delta$ polymorph suggesting that better performance can be achieved by
cycling Mg in the $\delta$ phase.
