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14
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85034871080
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(1983)
Physica
, vol.121B
, pp. 153
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24
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84950860547
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The derivation of the virial equation of state and its subsequent application to hard spheres in three dimensions is given in a number of standard texts. See, e.g. Ref. 1, pp. 40 and 52. The key step in the derivation, recognizing that in the hard-sphere limit the derivative of the Boltzmann factor reduces to a one-dimensional delta function at the contact radius, is independent of dimensionality. It is therefore straightforward to generalize the standard development to arbitrary dimensionality.
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26
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84927236391
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We have made a small [formula omitted] correction to these points, as in Refs. 21 and 22. Note that despite the fact that more recent and more extensive simulations have been performed (see, in particular, Ref. 4) this calculation remains the one ranging furthest in density over the fluid range.
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(1960)
J. Chem. Phys.
, vol.33
, pp. 1439
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Adler, B.J.1
Wainwright, T.E.2
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29
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84950928454
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Summarized by
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33
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84950751751
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The [formula omitted] limit requires some careful definition. From Eq. (2.2) it is clear that even the second virial coefficient b will vanish if the limit is taken so that [formula omitted] remains finite. However, for the purposes of this paper (as illustrated in Figs. 1 and especially 3) the natural definition is one that keeps bp finite. In this case one can show by using the appropriate d-dimensional generalization of bipolar coordinates that the third virial coefficient vanishes exponentially as [formula omitted] See also
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37
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84950755259
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The uncertainties quoted for s are simply statistical estimates based on the curve fitting of the log-log representations (Fig. 2). Additional uncertainties arise from the uncertainties in the values used for [formula omitted] Hence, any future modifications of [formula omitted] will necessitate revisions of s. We particularly note, in this connection
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40
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85034871489
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Reference 1. pp. 344 and 345.
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41
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85034869857
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Estimates of η range from [formula omitted] to [formula omitted] for three-dimensional percolation:
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45
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85034868193
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Note that our prediction for the asymptotic sign of c(r) is in direct disagreement with that in Ref. 32.
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46
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84950780926
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The general approach of first assuming a functional form for [formula omitted] outside the core and then determining the necessary parameters so as to provide both thermodynamic self-consistency and agreement with known results is called the Generalized Mean Spherical Approximation (GMSA). The GMSA has been applied to hard spheres by
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60
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85034880492
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Particularly worrisome is the small size of the sample used in Ref. 20 to prevent the spontaneous crystallization observed in the much larger simulation of Ref. 4. However, it may turn out to be impossible to satisfy simultaneously the requirements of low [formula omitted] large [formula omitted] and large numbers of particles.
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