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We evidence via a computation in the reciprocal space the asymptoticbehaviour of the high order virial coefficients for a hard sphere system. Thesecoefficients, 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 states ofthe hard sphere system at high density when the fluid phase is no longer thestable one; in the disordered phase this equation of states exhibits a simplepole at the random close packing density. We can then estimate the packingdensities of the freezing point of the disordered phase and also of the meltingpoint of the fcc ordered phase. The results are compared with those of thenumerical simulations.

Asymptotic expression of the virial coefficients for hard sphere systems