A remark on the Percus-Yevick approximation in higher dimensions

Freasier, B.C.; Isbister, Dennis J.

doi:10.1080/00268978100100711

Cited by 46 publications

(48 citation statements)

References 6 publications

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“…The exact solution for the PY equation in three, 16,[33][34][35][36] five, 16,37 and seven 38 dimensions is known. However, there is no closed form solution for even dimensions.…”

Section: ͑4͒mentioning

confidence: 99%

Structure factor for hard hyperspheres in higher dimensions

Whitlock

Bishop

Tiglias

2007

The Journal of Chemical Physics

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The structure factor for hard hyperspheres in two to eight dimensions is computed by Fourier transforming the pair correlation function obtained by computer simulation at a variety of densities. The resulting structure factors are compared to the known Percus-Yevick equations for odd dimensions and to the model proposed by Leutheusser [J. Chem. Phys. 84, 1050 (1986)] and Rosenfeld [J. Chem. Phys. 87, 4865 (1987)] in even dimensions. It is found that there is fine agreement among all these approaches at low to moderate densities but that the accuracy of the analytical models breaks down as the freezing transition is approached. The structure factor gives another insight into the decrease in the ordering of the hyperspheres as the dimension is increased.

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“…The exact solution for the PY equation in three, 16,[33][34][35][36] five, 16,37 and seven 38 dimensions is known. However, there is no closed form solution for even dimensions.…”

Section: ͑4͒mentioning

confidence: 99%

Structure factor for hard hyperspheres in higher dimensions

Whitlock

Bishop

Tiglias

2007

The Journal of Chemical Physics

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“…The basic quantity required for this implementation is the contact value of the RDF for a partially coupled particle, which corresponds to a binary mixture with one infinitely dilute component. Although the PY approximation had been solved for the 5D-sphere system by Freasier and Isbister [38] and by Leutheusser [42], their methods do not provide the RDF of a binary mixture required here. For this purpose we have used the RFA method [35] which exactly solves the PY equation for hypersphere mixtures in odd dimensionalities.…”

Section: Final Remarksmentioning

confidence: 99%

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

Rohrmann

Santos

2015

Phys. Rev. E

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We use the Percus-Yevick approach in the chemical-potential route to evaluate the equation of state of hard hyperspheres in five dimensions. The evaluation requires the derivation of an analytical expression for the contact value of the pair distribution function between particles of the bulk fluid and a solute particle with arbitrary size. The equation of state is compared with those obtained from the conventional virial and compressibility thermodynamic routes and the associated virial coefficients are computed. The pressure calculated from all routes is exact up to third density order, but it deviates with respect to simulation data as density increases, the compressibility and the chemical-potential routes exhibiting smaller deviations than the virial route. Accurate linear interpolations between the compressibility route and either the chemical-potential route or the virial one are constructed.

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“…(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. …”

Section: (13)mentioning

confidence: 99%

“…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]:…”

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

Santos

2012

Phys. Rev. Lett.

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The chemical potential of a hard-sphere fluid can be expressed in terms of the contact value of the radial distribution function of a solute particle with a diameter varying from zero to that of the solvent particles. Exploiting the explicit knowledge of such a contact value within the PercusYevick (PY) theory, and using standard thermodynamic relations, a hitherto unknown PY equation of state, p/ρkBT = −(9/η) ln(1 − η) − (16 − 31η)/2(1 − η) 2 , is unveiled. This equation of state turns out to be better than the one obtained from the conventional virial route. Interpolations between the chemical-potential and compressibility routes are shown to be more accurate than the widely used Carnahan-Starling equation of state. The extension to polydisperse hard-sphere systems is also presented. 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]:

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A remark on the Percus-Yevick approximation in higher dimensions

Cited by 46 publications

References 6 publications

Structure factor for hard hyperspheres in higher dimensions

Structure factor for hard hyperspheres in higher dimensions

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

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

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