  The hard disc system plays a fundamental role in the study of two-dimensional
matters [1-3]. High-precision compressibility data from computer simulations
have been reported for all the phases and phase transition regions [4-15]. In
particular, Bernard and Krauth (Phys. Rev. Lett., 107, 155704, 2011) [10]
presented a complete and accurate picture of the phase transitions of the hard
disc system with simulation results. However, thorough descriptions of the
system depend on analytical equations of state (EoS) over the entire density
range. While majority of EoS published are for the stable fluid region only
[1,16], few attempted the liquid-hexact transition region (Phys. Rev. Lett.,
11, 241, 1963 [17]; Phys. Rev. E. 63, 042201, 2001 [18]; 74, 061106, 2006
[19]). All the EoS currently available are incapable of quantitative
descriptions of the phase transitions. Here we construct a simple EoS to
reproduce high-precision simulation data for all the stable liquid,
liquid-hexatic transition region and hexatic phase. A global EoS is then
obtained when the new EoS is smoothly united with a revisited EoS for the solid
phase. Using this global equation, we are able to accurately identify all the
phases and the phase transitions from the stable liquid to hexatic, then to
solid phases. The liquid-hexatic transition is found to be of weak first-order,
namely discontinuous in density and the Gibbs free energy while continuous in
entropy and the Helmholtz free energy. The hexatic-solid transition is a
continuous high-order phase transition.
