Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

Clisby, Nathan; McCoy, Barry M.

doi:10.1007/s10955-005-8080-0

Cited by 217 publications

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“…As in a variety of interacting many-body systems [12], we expect studies of hard-sphere packings in high dimensions to yield great insight into the corresponding phenomena in lower dimensions. Analytical investigations of hard-spheres can be readily extended into arbitrary spatial dimension [13,14,15,16,17,20,21,22,23,24,25,26,27,28,29] and high dimensions can therefore be used as a stringent testing ground for such theories. Along these lines and of particular interest to this paper, predictions have been made about long-wavelength density fluctuations [25] and decorrelation [27,28] in disordered hard-sphere packings in high dimensions.…”

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 meaning of this is controversial and the present authors argue that the true large k behaviour is not seen in the first 10 coefficients. More complete analysis and discussion of this is given in [19].…”

Section: Resultsmentioning

confidence: 99%

“…Our results for the leading singularity on the real positive azis in dimensions D = 2, 3, 4 are tabulated in Table 3. More detailed analysis will appear in [19]. The notation L, M ; N refers to an inhomogeneous first order differential approximant, which is the solution of…”

Section: Differential Approximantsmentioning

confidence: 99%

“…In a series of papers [17,18,19] we have extended these numerical computations by computing virial coefficients up through B 10 in D = 2, 3, · · · , 8. We use the method of Ree-Hoover diagrams as evaluated by Monte Carlo integration.…”

Section: Introductionmentioning

confidence: 99%

“…We use the method of Ree-Hoover diagrams as evaluated by Monte Carlo integration. The details are given in [19] Our results are given in Table 1 in the form of B k /B k−1 2 where…”

Section: Introductionmentioning

confidence: 99%

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5As indicated in Chapter 1 , in liquids a natural reference state is absent because the structure is highly irregular. Therefore, in the description of liquids and solutions, we essentially will need statistical thermodynamics, which provides the link between the microscopic considerations and concepts to the macroscopic -in particular thermodynamic -aspects. In this chapter we introduce and outline statistical thermodynamics in a concise way. First, we deal with the basic formalism and apply that to noninteracting molecules. Thereafter, the effect of intermolecular interactions is introduced, leading to the virial description. While being a reasonable approach for low-and medium-density fl uids, the virial approach does not work for liquids. The reasons for this are briefl y discussed. Statistical Thermodynamics ⎞ ⎠ ⎟ +…At high temperature the summation can be approximated by integration and thus

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Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

Cited by 217 publications

References 81 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

The Transition from Microscopic to Macroscopic: Statistical Thermodynamics

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