  We report extensive simulation studies of phase behaviour in single component
systems of particles interacting via a core-softened interparticle potential.
Two recently proposed examples of such potentials are considered; one in which
the hard core exhibits a shoulder, (Sadr-Lahijany et al, Phys. Rev. Lett. 81,
4895 (1998)) and the other in which the softening takes the form of a linear
ramp (Jagla, Phys. Rev. E63, 061501 (2001)). Using a combination of
state-of-the-art Monte Carlo methods, we obtain the gas, liquid and solid phase
behaviour of the shoulder model in two dimensions. We then focus on the
thermodynamic anomalies of the liquid phase, namely maxima in the density and
compressibility as a function of temperature. Analysis of the finite-size
behaviour of these maxima suggests that, rather than stemming from a metastable
liquid-liquid critical point, as previously supposed, they are actually induced
by the quasi-continuous nature of the two dimensional freezing transition. For
the ramp model in three dimensions, we confirm the existence of a stable
liquid-liquid ("second") critical point occurring at higher pressure and lower
temperature than the liquid-gas critical point. Both these critical points and
portions of their associated coexistence curves are located to high precision.
In contrast to the shoulder model, the observed thermodynamic anomalies of this
model are found to be authentic, i.e. they are not engendered by an incipient
new phase. We trace the locus of density and compressibility maxima, the former
of which appears to converge on the second critical point.
