Asymptotic expression of the virial coefficients for hard sphere systems

We evidence via a computation in the reciprocal space the asymptotic behavior of the high order virial coefficients for a hard sphere system. These coefficients, if their order is high enough, are those of a geometric series. We thus are able to give an explicit expression of the equation of state of the hard sphere system at high density when the fluid phase is no longer the stable one; in the disordered phase this equation of state exhibits a simple pole at the random close packing density. We can then estimate the packing densities of the freezing point of the disordered phase and also of the melting point of the fcc ordered phase. The results are compared with those of the numerical simulations.