Phase Behaviour of Purely Repulsive Systems: Violation of Traditional van der Waals Picture

Several purely repulsive potentials (PRP) are investigated theoretically. In contrast to the traditional van der Waals picture, it is found that normal gas--liquid transition emerges only on condition that the PRP as a function of particle separation holds a discontinuous point, or an indifferentiable point,
or is differentiable but with an additional length scale besides the hard sphere diameter.