Virial series for fluids of hard hyperspheres in odd dimensions

Abstract

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E 76, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation for a fluid of hard spheres in (odd) d dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients B(j) are expressed in terms of the solution of a set of (d-1)/2 coupled algebraic equations which become nonlinear for d>/=5. Results have been derived up to d=13. A confirmation of the alternating character of the series for d>/=5, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension 2eta(1/d), where eta is the packing fraction, is wholly consistent with the limiting value of 1 for d-->infinity. Finally, the values for B(j) predicted by the virial and compressibility routes in the Percus-Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. 122, 15 (2006)].

A branch-point approximant for the equation of state of hard spheres

“…37,38 It is very likely that these features are not artifacts of the PY approximation but would be shared by the exact EOSs. However, in the case of hard spheres ͑d =3͒, the radius of convergence of the PY EOS is artificially = 1 and, as stated above, there is no definite indication about the nature of the singularity responsible for the true radius of convergence or its value.…”

Solution of the Percus–Yevick equation for hard hyperspheres in even dimensions

“…The study of such systems has played a central role in the understanding of classical fluids and serves as a starting point for the construction of perturbation theories of fluid properties. A particular interest has been systems of hard hyperspheres 1,2,3,4,5,6,7,8,9,10,11,12,13 (i.e. the generalization of spheres to dimensions larger than three, d > 3).…”

“…On the other hand, approximate methods have led to very good predictions in the low density phase. Among the most widely used approximations is the Percus-Yevick (PY) equation for d-dimensional hard spheres 18 , which is exact to first order in the density of the fluid 11 , ρ. A great deal of progress has been made towards understanding the solutions of the PY equation in odd dimensions.…”

Equation of state for five-dimensional hyperspheres from the chemical-potential route

“…As a peculiar feature, it may be observed that, in contrast to the three-dimensional (3D) case [8], the µ route yields irrational virial coefficients (due to the logarithmic term) for n ≥ 5. On the other hand, the virial and compressibility routes yield rational numbers for the virial coefficients of hypersphere systems in all odd dimensions [39]. Table II shows the three sets of PY virial coefficients in numerical format and compares them with the exact result for n = 4 [23] and with recent accurate values for 5 ≤ n ≤ 12 [34].…”

“…It is interesting to remark that the three PY routes capture the alternating sign change between n = 7 and n = 12, while a negative sign of b 6 is wrongly anticipated by the virial and µ routes. In the PY case, the alternating character is related to the existence of a branch point singularity on the negative real axis (at η = −3/2 + 5 √ 3/6 ≃ −0.0566), which determines the radius of convergence of the virial series [39].…”