Structure of hard-hypersphere fluids in odd dimensions

Abstract

The structural properties of single component fluids of hard hyperspheres in odd space dimensionalities d are studied with an analytical approximation method that generalizes the rational function approximation earlier introduced in the study of hard-sphere fluids [S. B. Yuste and A. Santos, Phys. Rev. A 43, 5418 (1991)]. The theory makes use of the exact form of the radial distribution function to first order in density and extends it to finite density by assuming a rational form for a function defined in Laplace space, the coefficients being determined by simple physical requirements. Fourier transform in terms of reverse Bessel polynomials constitute the mathematical framework of this approximation, from which an analytical expression for the static structure factor is obtained. In its most elementary form, the method recovers the solution of the Percus-Yevick closure to the Ornstein-Zernike equation for hyperspheres at odd dimensions. The present formalism allows one to go beyond by yielding solutions with thermodynamic consistency between the virial and compressibility routes to any desired equation of state. Excellent agreement with available computer simulation data at d=5 and d=7 is obtained.

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

“…Since hard spheres are athermal, the energy route to the EOS becomes useless. In order to evaluate g 12 (σ + 12 ) within the RFA approach [35], it is convenient to introduce the Laplace functional defined by 6) where g ij (r) is the RDF of the pair (i, j) and θ n (sr) is the reverse Bessel polynomial of order n = (d − 3)/2 [27]. This functional is directly related to the static structure factors S ij (k) of a multicomponent fluid,…”

“…Besides the intrinsically interesting properties of hard particle systems, they are an important basis for constructing more complicate models, so that there are active theoretical efforts to study them in dimensions d > 3 (see, for instance, Refs. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and references therein).…”

Chemical-Potential Route: A Hidden Percus-Yevick Equation of State for Hard Spheres

“…(29) for more general systems can be very helpful for the construction of accurate EOS. Extensions of this work to sticky hard spheres [30] and to hyperspheres [9,10,13] are planned. …”

“…Motivation and discussion.-As is well known, the hard-sphere (HS) model is of great importance in condensed matter, colloids science, and liquid state theory from both academic and practical points of view [1][2][3]. The model has also attracted a lot of interest because it provides a nice example of the rare existence of nontrivial exact solutions of an integral-equation theory, namely the Percus-Yevick (PY) theory [4] for odd dimensions [5][6][7][8][9][10][11][12][13][14][15].As generally expected from an approximate theory, the radial distribution function (RDF) provided by the PY integral equation suffers from thermodynamic inconsistencies; i.e., the thermodynamic quantities derived from the same RDF via different routes are not necessarily mutually consistent. In particular, the PY solution for three-dimensional one-component HSs of diameter σ yields the following expression for the compressibility factor Z ≡ p/ρk B T (where p is the pressure, ρ is the number density, k B is Boltzmann's constant, and T is the temperature) through the virial (or pressure) route [5][6][7]:…”

“…The model has also attracted a lot of interest because it provides a nice example of the rare existence of nontrivial exact solutions of an integral-equation theory, namely the Percus-Yevick (PY) theory [4] for odd dimensions [5][6][7][8][9][10][11][12][13][14][15].…”