The Fourth Virial Coefficient of a Fluid of Hard Spheres in Odd Dimensions

Lyberg, I.

doi:10.1007/s10955-005-3020-6

Cited by 38 publications

(28 citation statements)

References 19 publications

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“…For the low-density fluid, lower-order virial coefficients, B 2 , B 3 , and B 4 , are known exactly for arbitrary dimensionality [13,14,17]. Higher-order virial coefficients have been calculated by Monte Carlo simulation, B 5 , B 6 and B 7 for both d = 4 and d = 5 [16] and B 8 for d = 4 [16], and analytically [18,19]. The pair correlation function for equilibrium fluids has been studied and a decrease in ordering with increasing dimension was readily apparent [32].…”

Section: Introductionmentioning

confidence: 99%

Packing hyperspheres in high-dimensional Euclidean spaces

et al. 2006

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We present the first study of disordered jammed hard-sphere packings in four-, five-and sixdimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d = 4, 5 and 6 to be φ M RJ ≃ 0.46, 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling formand c 2 = 2.56, which appears to be consistent with high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g 2 (r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed "decorrelation principle," which, among othe things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d → ∞. As in three dimensions (where φ M RJ ≃ 0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g 2 (r) at contact with power-law exponent ≃ 0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g 2 (r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, φ F ≃ 0.32 and φ M ≃ 0.39, respectively, for d = 4, and φ F ≃ 0.19 and φ M ≃ 0.24, respectively, for d = 5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension. * Electronic address: torquato@electron.princeton.edu 2

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Section: Introductionmentioning

confidence: 99%

Packing hyperspheres in high-dimensional Euclidean spaces

et al. 2006

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“…The problem was first studied analytically by by van der Waals [1], Boltzmann [2], and van Laar [3] who computed the coefficients up through B 4 . The computation of B 4 for D = 2 was first done in 1964 by Rowlinson [4] and Hemmer [5] and very recently these analytic computations for B 4 have been extended to D = 4, 6, 8, 10, 12 by the present authors [6], and by Lyberg [7] for D = 5, 7, 9, 11. All other computations for the hard sphere gas are by means of computer.…”

Section: Introductionmentioning

confidence: 87%

New results for virial coefficients of hard spheres inD dimensions

Clisby

McCoy

2005

Pramana - J Phys

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“…3,5,17,29,44,48,53 The extension to arbitrary dimensionality of this hard spherical particle system has been the object of several studies too. 7,14,15,27,31,32,54 Although the apparent simplicity of these systems, only a few exact analytical results for the homogeneous system are known concerning mainly the dimensional dependence of the first four virial coefficients 14,30,32 existing numerical extensions to higher order. 15,25 Exact virial series studies has been also done in inhomogeneous systems 8,34,45 in three dimensions.…”

Section: Introductionmentioning

confidence: 99%

Statistical mechanics of two hard spheres in a spherical pore, exact analytic results in D dimension

Urrutia

Szybisz

2010

Journal of Mathematical Physics

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Structures of hard-sphere fluids from a modified fundamental-measure theory J. Chem. Phys. 117, 10156 (2002) This work is devoted to the exact statistical mechanics treatment of simple inhomogeneous few-body systems. The system of two hard spheres ͑HSs͒ confined in a hard spherical pore is systematically analyzed in terms of its dimensionality D. The canonical partition function and the one-and two-body distribution functions are analytically evaluated and a scheme of iterative construction of the D + 1 system properties is presented. We analyze in detail both the effect of high confinement, when particles become caged, and the low density limit. Other confinement situations are also studied analytically and several relations between the two HSs in a spherical pore, two sticked HSs in a spherical pore, and two HSs on a spherical surface partition functions are traced. These relations make meaningful the limiting caging and low density behavior. Turning to the system of two HSs in a spherical pore, we also analytically evaluate the pressure tensor. The thermodynamic properties of the system are discussed. To accomplish this statement we purposely focus in the overall characteristics of the inhomogeneous fluid system, instead of concentrate in the peculiarities of a few-body system. Hence, we analyze the equation of state, the pressure at the wall, and the fluid-substrate surface tension. The consequences of new results about the spherically confined system of two HSs in D dimension on the confined many HS system are investigated. New constant coefficients involved in the low density limit properties of the open and closed systems of many HS in a spherical pore are obtained for arbitrary D. The complementary system of many HS which surrounds a HS ͑a cavity inside of a bulk HS system͒ is also discussed.

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The Fourth Virial Coefficient of a Fluid of Hard Spheres in Odd Dimensions

Abstract: The fourth virial coefficient is calculated exactly for a fluid of hard spheres in odd dimensions up to 11.

Cited by 38 publications

References 19 publications

Packing hyperspheres in high-dimensional Euclidean spaces

Packing hyperspheres in high-dimensional Euclidean spaces

New results for virial coefficients of hard spheres inD dimensions

Statistical mechanics of two hard spheres in a spherical pore, exact analytic results in D dimension

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