  We investigate theoretically the freezing behaviour of a two-dimensional (2D)
system of hard discs on a one-dimensional (1D) external potential (typically
called laser-induced freezing). As shown by earlier theoretical and numerical
studies, one observes freezing of the modulated liquid upon increase of the
substrate potential amplitude, and re-entrant melting back into the modulated
liquid when the substrate potential amplitude is increased even further. The
purpose of our present work is to calculate the freezing and re-entrant melting
phase diagram based on information from the bulk system. To this end, we employ
an integrated pressure-balance equation derived from density functional theory
[Phys. Rev. E \textbf{101}, 012609 (2020)]. Furthermore, we define a measure to
quantify the influence of registration effects that qualitatively explain
re-entrant melting. Despite severe approximations, the calculated phase diagram
shows good agreement with the known phase diagram obtained by Monte Carlo
simulations.
