Equations of state for D-dimensional hard sphere fluids

An equation of state à la Carnahan–Starling for a five-dimensional fluid of hard hyperspheres

“…Although not present in nature, fluids of hard hyperspheres in high dimensions (d ≥ 4) have attracted the attention of a number of researchers over the last twenty years. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] Among these studies, one of the most important outcomes was the realization by Freasier and Isbister 1 and, independently, by Leutheusser 4 that the Percus-Yevick (PY) equation 15 admits an exact solution for a system of hard spheres in d = odd dimensions. In the special case of a five-dimensional system (d = 5), the virial series representation of the compressibility factor Z ≡ p/ρk B T (where p is the pressure, ρ is the number density, k B is the Boltzmann constant, and T is the temperature) is Z(η) = ∞ n=0 b n+1 η n , where η = (π 2 /60)ρσ 5 is the volume fraction (σ being the diameter of a sphere One of the simplest proposals is Song, Mason, and Stratt's (SMS), 7 who, by viewing the Carnahan-Starling (CS) EOS for d = 3 16 as arising from a kind of meanfield theory, arrived at a generalization for d dimensions that makes use of the first three virial coefficients.…”

“…Baus and Colot (BC) 6 proposed a rescaled (truncated) virial expansion that explicitly accounts for the first four virial coefficients. A slightly more sophisticated EOS is the rescaled Padé approximant proposed by Maeso et al (MSAV), 12 which reads…”

Equation of State of Four- and Five-Dimensional Hard-Hypersphere Mixtures

“…The interest in studying systems of d -dimensional hard spheres has been present for many decades and still continues to stimulate intensive research [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 ,…”

“…While of course real experiments cannot be performed in these systems, they are amenable to computer simulations and theoretical developments. Many aspects concerning hard hyperspheres have been already dealt with, such as thermodynamic and structural properties [ 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , …”

Monte Carlo simulation for a gas of hard spheres in d-dimensional space: Equilibrium structure and state equations

“…A remarkable number of theoretical and simulation results has typically been calculated for the HS system in two and three dimensions [8]. A straightforward extension of the system consists of studying the properties of a gas in high dimensions, with a Euclidean metric space [5][6][7][8][9][10][11]. These studies are relevant not only from a theoretical perspective, since for some complex physical systems it has provided a better understanding of the equilibrium states [12][13][14][15][16][17][18][19].…”