Multiple liquid-liquid critical points and anomalies in core-softened   potentials
  The relation between liquid-liquid phase transitions and waterlike density anomalies in core-softened potentials of fluids was investigated in an exactly solvable one dimensional lattice model and a in a three dimensional fluid with fermi-like potential, the latter by molecular dynamics. Both systems were shown to present three liquid phases, two liquid-liquid phase transitions closely connected to two distinct regions of anomalous density increase. We propose that an oscillatory behavior observed on the thermal expansion coefficient as a function of pressure can be used as a signature of the connection between liquid-liquid phase and density. 
Marco Aur\'elio A. Barbosa, Evy Salcedo and Marcia Barbosa
cond-mat.soft