  Monte Carlo simulations within the grand canonical ensemble are used to
explore the liquid-vapour coexistence curve and critical point properties of
the Lennard-Jones fluid. Attention is focused on the joint distribution of
density and energy fluctuations at coexistence. In the vicinity of the critical
point, this distribution is analysed using mixed-field finite-size scaling
techniques aided by histogram reweighting methods. The analysis yields highly
accurate estimates of the critical point parameters, as well as exposing the
size and character of corrections to scaling. In the sub-critical coexistence
region the density distribution is obtained by combining multicanonical
simulations with histogram reweighting techniques. It is demonstrated that this
procedure permits an efficient and accurate mapping of the coexistence curve,
even deep within the two phase region.
