  The effect of polydispersity on the phase diagram of a simple binary mixture
is to split the binodal curve into cloud and shadow curves that cross at the
critical point (which, in general, is not at the maximum of either curve).
Recent theories of polydispersity have shown, in favorable cases, how to
project the (infinite-dimensional) free energy of the polydisperse system into
a low dimensional space of `moment densities'. We address here the issue of how
to construct a Landau expansion from the projected free energy. For the
simplest case where the excess free energy depends on one moment density
$\rho_1$ (this includes Flory Huggins theory for length-polydisperse chains) we
show that the minimal expansion remains quartic in $\rho_1$ but nonetheless has
seven independent coefficients, not two. When this expansion is handled
correctly all the peculiar qualitative features of the polydisperse phase
diagram are recovered, as are the slopes of the cloud and shadow curves, and
the curvature of the cloud. However, to correctly predict the curvature of the
shadow, certain fifth order terms must be retained. We also consider the phase
diagram on the temperature-pressure plane, where the coexistence line broadens
into a region. In general, the critical point lies between the point of maximum
temperature and the point of maximum pressure on the perimeter of this region.
This behaviour is again captured by the Landau expansion.
