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 {\bf 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 dimension. 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. As a byproduct of this study, an exact and explicit polynomial expression for the intersection volume of two identical hyperspheres in arbitrary odd dimensions is given.

18 pages, 7 figures; v2: new references added plus minor changes; to be published in PRE

References in corpus (5)

• Packing Hyperspheres in High-Dimensional Euclidean Spaces
• Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids
• Low-temperature and high-temperature approximations for penetrable-sphere fluids. Comparison with Monte Carlo simulations and integral equation theories
• Percus-Yevick theory for the structural properties of the seven-dimensional hard-sphere fluid
• Demixing can occur in binary hard-sphere mixtures with negative non-additivity

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• Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation
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• Solution of the Percus-Yevick equation for hard discs
• Structural and Thermodynamic Properties of Hard-Sphere Fluids
• Virial series for fluids of hard hyperspheres in odd dimensions
• Effect of Dimensionality on the Percolation Thresholds of Various $d$-Dimensional Lattices
• Chemical-Potential Route: A Hidden Percus-Yevick Equation of State for Hard Spheres
• Multicomponent fluids of hard hyperspheres in odd dimensions
• Random Close Packing and the Hard Sphere Percus-Yevick Theory
• Fluid-solid transition in hard hyper-sphere systems
• Exact solution of the Percus-Yevick integral equation for fluid mixtures of hard hyperspheres
• Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation
• Thermodynamics and structure of simple liquids in the hyperbolic plane
• Density functional for hard hyperspheres from a tensorial-diagrammatic series
• Dynamical density functional theory for the diffusion of injected Brownian particles
• Equation of State of Four- and Five-Dimensional Hard-Hypersphere Mixtures
• Liquid pair correlations in four spatial dimensions: Theory versus simulation
• Fluid-fluid versus fluid-solid demixing in mixtures of parallel hard hypercubes
• A Mechanical Approach to One-Dimensional Interacting Gas
• Temperature Expansions in the Square-Shoulder Fluid I: the Wiener-Hopf Function
• Dynamic Space Packing
• Equation of state for five-dimensional hyperspheres from the chemical-potential route