Third and fourth virial coefficients of hard hyperspheres of arbitrary dimensionality

We derive exact expressions for the third virial coefficient B3 and two of the three terms B4(⧠) and B4(⧅) contributing to the fourth virial coefficient B4 for an assembly of hard hyperspheres of arbitrary dimensionality d. A semiempirical formula, accurate for all d, is proposed for the remaining term B4(⊠). This formula is suggested by the large-d asymptotic behavior of B3 and B4(⧠). We find that B4 is positive for d<d4 and negative for d>d4, with d4 estimated to be d4≈7.8, whereas B4(⧠)+B4(⧅) changes sign for d = 9.073