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Advanced Statistical Mechanics$

Barry M McCoy

Print publication date: 2009

Print ISBN-13: 9780199556632

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199556632.001.0001

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Ree–Hoover virial expansion and hard particles

Ree–Hoover virial expansion and hard particles

Chapter:
(p.181) 7 Ree–Hoover virial expansion and hard particles
Source:
Advanced Statistical Mechanics
Author(s):

Barry M. McCoy

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199556632.003.0007

Abstract and Keywords

This chapter derives the modification of the Mayer expansion made by Ree and Hoover. Analytic expressions for the virial coefficients B2,B3, and B4 are given and Monte–Carlo results for Bn for 5 ≤ n ≤ 10 in dimensions 1 ≤ D ≤ 10 are presented. Various approximate equations of state used to ‘fit’ these coefficients are summarized. Low order virial coefficients for hard squares, cubes and hexagons are given. Open questions relating to the signs of the virial coefficients for hard spheres and discs and to the relation of virial expansions to freezing are discussed.

Keywords:   hard discs, hard spheres, hard hexagons, Monte–Carlo evaluations, hard cubes, hard squares

In this chapter we study what may be considered to be the simplest potential in classical statistical mechanics, hard spheres in D dimensions defined by the potential

(7.1)                       Ree–Hoover virial expansion and hard particles

Equation (7.1) says that the centers of the two spheres cannot be closer than σ which is traditionally called the hard core diameter of the sphere.

The potential (7.1) is particularly simple because the Boltzmann weights are independent of temperature. Thus P/kBT is a function of density alone and the internal energy is the same as the perfect gas. There is no energy scale in the problem and all the physics is determined by entropy, geometry and combinatorics. What we learn from the study of hard spheres is basic to what has become known in recent years as “soft condensed matter physics”.

The study of the virial coefficients of the hard sphere gas splits into two separate parts: the generation of the graphs and the evaluation of the integrals. The generation of the Mayer graphs is a serious problem because the number of these graphs grows very rapidly. The computation of integrals is difficult because there are strong cancellations between graphs, some of which are positive and some of which are negative.

A considerable simplification in the computation of virial coefficients was made in the mid 1960s by Ree and Hoover [13] who introduced a rearrangement of the Mayer graphs which decreases the number of graphs. We present this expansion in section 7.1. For the hard core gas (7.1) the Ree–Hoover expansion has the additional property that when the virial coefficient Bk is evaluated in D dimensions if k − 3 ≥ D then some diagrams vanish for the hard core potential which do not vanish in general, diagrams This vanishing is vividly demonstrated in section 7.2 where we consider the very simple case of the Tonks gas [4], which is the one-dimensional case of the hard sphere gas (7.1), and show that there is exactly one nonvanishing Ree-Hoover graph for each virial coefficient.

In section 7.3 we analytically evaluate the virial coefficients for hard spheres B 2, B 3, and B 4 [513]. The results are given in Tables 7.5 and 7.6. The derivations for B 2 and B 3 are given in detail. The virial coefficients B 5B 10 have been evaluated numerically [1, 3, 1417]. We discuss these computations in section 7.4 and tabulate the results in Table 7.8.

(p.182) In section 7.5 we discuss the possible behavior of the hard sphere virial coefficients for k ≥ 11 and examine the behavior of certain graphs for large values of k and in section 7.6 we use the results to estimate the radius of convergence of the virial series and discuss the location of the leading singularity in the complex density plane. We also present various of the approximate equations of state that have been proposed for hard spheres in three dimensions [1826].

The hard sphere potential (7.1) is radially symmetric and the spatial variable r is in the continuum. However, the Mayer and Ree–Hoover expansions are equally valid for potentials which have an angular dependence and on a lattice the analogue of (7.1) is the pair potential U(r kr k) for a

lattice gas with nearest neighbor exclusion

(7.2)                       Ree–Hoover virial expansion and hard particles

In section 7.7 we present the known results in the continuum for parallel hard squares and cubes [27] and the lattice gas results for hard squares on a square lattice [28, 29], hard hexagons on the triangular lattice [2933] and hard particles with nearest neighbor exclusion on the cubic, fcc and bcc lattices in three dimensions [30].

In section 7.8 we consider nonspherical convex bodies which are allowed to rotate. In three dimensions the first computation of a second virial coefficient for nonspherical shapes was given by Onsager in 1949 [34], and in two dimensions B 2, B 3 and B 4 have been computed for ellipses, rectangles and needles [35, 36].

We conclude in section 7.9 with a summary of some of the open questions in the study of virial expansions of hard particles.

7.1 The Ree–Hoover expansion

In the previous chapter we showed that the coefficients in the virial expansion of the pressure

(7.3)                       Ree–Hoover virial expansion and hard particles
are
(7.4)                       Ree–Hoover virial expansion and hard particles
where Vk(r 1, ⃛ , r k) is the sum of all numbered biconnected Mayer graphs with k points. In these Mayer graphs each line stands for the factor
(7.5)                       Ree–Hoover virial expansion and hard particles

The numbered Mayer graphs of a certain type are obtained by labeling the vertices of the basic unnumbered graph in all possible distinct ways. Calling si[n] the number (p.183) of labelings of the graph of type i with n points and calling Si[n] the integral of the unlabeled graph of type i and n points we have

(7.6)                       Ree–Hoover virial expansion and hard particles

In 1964 Ree and Hoover in [13] introduced a simple and very useful modification of this expansion by introducing in addition to (7.5) the function

(7.7)                       Ree–Hoover virial expansion and hard particles
with the property that
(7.8)                       Ree–Hoover virial expansion and hard particles

The basic idea of the re-expansion is to note that, in a general Mayer graph, points are either connected by a line which contributes a factor of f or they are not connected. Every pair of points which is not connected can be thought of as contributing a factor of 1 to the integral. The Ree–Hoover expansion is to replace this 1 by                       Ree–Hoover virial expansion and hard particles and to rewrite the expansion in terms of integrals where all points are connected by bonds which now are either f or                       Ree–Hoover virial expansion and hard particles.

As a first example of the utility of this method, consider the fourth virial coefficient. There are three contributing graphs as shown in Fig. 7.1 and the symmetry numbers are

(7.9)                       Ree–Hoover virial expansion and hard particles

                      Ree–Hoover virial expansion and hard particles

Fig. 7.1 The three unlabeled Mayer diagrams which contribute to B 4 and their symmetry numbers si[4].

Thus

(7.10)                       Ree–Hoover virial expansion and hard particles
with
(7.11)                       Ree–Hoover virial expansion and hard particles
(7.12)                       Ree–Hoover virial expansion and hard particles
(p.184)
(7.13)                       Ree–Hoover virial expansion and hard particles

Now in S 3[4] we insert the factors

(7.14)                       Ree–Hoover virial expansion and hard particles
and in S 2[4] we insert the factor
(7.15)                       Ree–Hoover virial expansion and hard particles
and find that
(7.16)                       Ree–Hoover virial expansion and hard particles
where
(7.17)                       Ree–Hoover virial expansion and hard particles
(7.18)                       Ree–Hoover virial expansion and hard particles

Hence we have reduced the number of diagrams to be considered from three to two. The computation is summarized graphically in Fig. 7.2 where the dotted lines represent the factor                       Ree–Hoover virial expansion and hard particles.

                      Ree–Hoover virial expansion and hard particles

Fig. 7.2 The graphical representation of the reduction of the three Mayer diagrams for B 4 to the two Ree–Hoover diagrams. The dotted lines represent the factors .

Thus we have the final result

(7.19)                       Ree–Hoover virial expansion and hard particles

To proceed further in a systematic fashion we follow [2] and denote the combinatorial factor in the Ree–Hoover expansion for the Ree–Hoover integral of k points (p.185)                       Ree–Hoover virial expansion and hard particles as                       Ree–Hoover virial expansion and hard particles. We refer to                       Ree–Hoover virial expansion and hard particles as the star content of the k point graph of type i. By definition

(7.20)                       Ree–Hoover virial expansion and hard particles

To compute                       Ree–Hoover virial expansion and hard particles, consider a Ree–Hoover diagram                       Ree–Hoover virial expansion and hard particles. This diagram is produced by expanding all those Mayer diagrams whose f functions are a subset of the f functions in                       Ree–Hoover virial expansion and hard particles. Denote those contributing diagrams as S l[j, k] and denote by Δfl the number of f bonds in                       Ree–Hoover virial expansion and hard particles that are not in the Mayer diagram Sl[j, k]. It is clear that the Sl[j, k] are exactly those diagrams which can be formed by removing Δfl f functions from the f functions in                       Ree–Hoover virial expansion and hard particles. Therefore we see that

(7.21)                       Ree–Hoover virial expansion and hard particles
where the minus sign appears because the expansion of                       Ree–Hoover virial expansion and hard particles introduces a minus sign with each of the f functions.

The equation (7.21) can be expressed by the following rule [2, p. 1637]:

Count the number of labeled Mayer graphs which can be formed by successively removing 0, 2, ⃛ of the f functions from the Ree–Hoover diagram                       Ree–Hoover virial expansion and hard particles and subtract from that the number of labeled Mayer graphs that can be formed from removing 1, 3, ⃛ of the f functions from                       Ree–Hoover virial expansion and hard particles. The resulting number (which can be positive, negative or zero) is the star content                       Ree–Hoover virial expansion and hard particles.

There is a useful property of the star content                       Ree–Hoover virial expansion and hard particles which helps in the computations.

Namely if                       Ree–Hoover virial expansion and hard particles and                       Ree–Hoover virial expansion and hard particles have the same type of                       Ree–Hoover virial expansion and hard particles bonds and differ only in the f bonds then

(7.22)                       Ree–Hoover virial expansion and hard particles

From this we find recursively that

(7.23)                       Ree–Hoover virial expansion and hard particles
where m is the smallest total number of points possible for the given configuration of                       Ree–Hoover virial expansion and hard particles bonds.

From (7.23) we find that for the “complete Ree–Hoover star diagram” which contains no bonds                       Ree–Hoover virial expansion and hard particles we have m = 2 and                       Ree–Hoover virial expansion and hard particles and thus we find for the star content

(7.24)                       Ree–Hoover virial expansion and hard particles

For B 5 and B 6 we explicitly give all contributing Ree-Hoover graphs in Tables 7.1 and 7.2. Here we use the notation that the diagram specified by Bk[m, i] has k points, and m points are connected by                       Ree–Hoover virial expansion and hard particles bonds. We list separately the factors sk[m, i].                       Ree–Hoover virial expansion and hard particles and the product

(7.25)                       Ree–Hoover virial expansion and hard particles
We also specify the graphs in two different notations either by giving the                       Ree–Hoover virial expansion and hard particles or the f bonds.

(p.186)

Table 7.1 Ree–Hoover diagrams for B 5. For each diagram we give the values of the combinatorial factor sk[m, i], the star content ãk[m, i] and the product                       Ree–Hoover virial expansion and hard particles where m is the number of points conected by                       Ree–Hoover virial expansion and hard particles bonds.

                      Ree–Hoover virial expansion and hard particles

The Ree–Hoover expansion has two definite advantages over the Mayer expansion. The first is that for any potential the number of Ree–Hoover graphs is smaller than the number of Mayer graphs. The second advantage is that for low dimensions certain diagrams vanish because of geometrical constraints. This effect is seen by examining B 5[5, 3] where in D = 2 (but not for D ≥ 3) it is impossible to find any configuration which satisfies both the restrictions that f(r) = 0 for                       Ree–Hoover virial expansion and hard particles and                       Ree–Hoover virial expansion and hard particles.

The number of contributing diagrams for general potentials, hard disks and hard spheres is given in Table 7.3 up through B 10. The values of these virial coefficients will be computed in section 7.3 and 7.4.

7.2 The Tonks Gas

The Tonks gas [4] is the name given to the particularly simple case of the hard sphere gas (7.1) in one dimension. In this case the partition function is particularly easy to compute if we note that a collision of hard rods in one dimension in a volume L behaves kinematically in the same way as free particles which move in volume of LNa where N is the number of rods. Thus if we replace V by LNa in the equation of state of the free gas we have the equation of state of the Tonks gas

(7.26)                       Ree–Hoover virial expansion and hard particles

When this is rewritten in the form of the virial expansion (7.3) we find

(7.27)                       Ree–Hoover virial expansion and hard particles
and thus
(7.28)                       Ree–Hoover virial expansion and hard particles
which is to be compared with the bound (6.40)
(7.29)                       Ree–Hoover virial expansion and hard particles

(p.187)

Table 7.2 Ree–Hoover diagrams for B 6. For each diagram we give the values of the combinatorial factor sk[m, i], the star content ãk[m, i] and the product                       Ree–Hoover virial expansion and hard particles where M is the nmber of points connected by                       Ree–Hoover virial expansion and hard particles bonds.

                      Ree–Hoover virial expansion and hard particles

(p.188)

Table 7.3 The number of Mayer and Ree–Hoover graphs which contribute to the virial coefficients up to order 10 from [17]. Some of the entries are only lower bounds because it is numerically difficult at times to distinguish between graphs which are very small and those which vanish identically.

                      Ree–Hoover virial expansion and hard particles

The Ree–Hoover expansion provides an exceptionally simple derivation of (7.27) because it is easily seen that in the evaluation of Bk all graphs vanish except the one in which each vertex is connected with every other vertex by the bond f i, j. Therefore we find from (7.20) and (7.24)

(7.30)                       Ree–Hoover virial expansion and hard particles
where, to obtain the last line, we have used the fact that the product contains k(k−1)/2 terms which are either −1 or zero. If we order the coordinates xj as x 1 < x 2 < ⃛ < xk and set x 1 = 0 by convention we see that (7.30) consists of k! integration regions all of which contribute equally. Thus we obtain
(7.31)                       Ree–Hoover virial expansion and hard particles
which agrees with (7.28).

It is also instructive to obtain the cluster integrals bk. We first use (6.128) to find

(7.32)                       Ree–Hoover virial expansion and hard particles

Then noting that

(7.33)                       Ree–Hoover virial expansion and hard particles
(p.189) we have
(7.34)                       Ree–Hoover virial expansion and hard particles

We now use Bürmann's theorem (6.217) with P as the variable z and set

(7.35)                       Ree–Hoover virial expansion and hard particles
where
(7.36)                       Ree–Hoover virial expansion and hard particles
to obtain
(7.37)                       Ree–Hoover virial expansion and hard particles

Thus comparing with (6.5) we find

(7.38)                       Ree–Hoover virial expansion and hard particles
from which we find b 2 = −σ and thus
(7.39)                       Ree–Hoover virial expansion and hard particles
which should be compared with Groeneveld's bounds (6.23).

7.3 Hard sphere virial coefficients B 2B 4 in two and higher dimensions

We now turn to the evaluation of Bk for dimensions D ≥ 2. The chronology of the computations is given in Table 7.4

The results for B 2, B 3 and B 4 are given in Tables 7.5 and 7.6. We will here derive the results for B 2 and B 3. The derivation of the results for B 4 is somewhat tedious and we refer the reader to the original papers.

7.3.1 Evaluation of B 2

The second virial coefficient for the hard sphere potential (7.1) is

(7.40)                       Ree–Hoover virial expansion and hard particles
where
(7.41)                       Ree–Hoover virial expansion and hard particles
and therefore
(7.42)                       Ree–Hoover virial expansion and hard particles
where VD(σ) is the volume of a sphere of radius σ in dimension D.

(p.190)

Table 7.4 Chronology of analytic computations of the hard sphere virial coefficients B 3 and B 4.

Date

Author (s)

Property

1899

Boltzmann [5]

B 3 for D = 3

1899

Boltzmann [5],van Laar [6]

B 4 for D = 3

1936

Tonks [4]

B 3 for D = 2

1951

Nijboer, van Hove [7]

Two center evaluation of B 4 in D = 3

1964

Rowlinson [8], Hemmer [9]

B 4 for D = 2

1982

Luban, Barum [10]

B 3 for arbitrary D

2003

Clisby, McCoy [11]

B 4 for D = 4, 6, 8, 10, 12

2005

Lyberg [12]

B 4 for D = 5, 7, 9, 11

One way to evaluate VD(r) is to note that (from dimensional considerations)

(7.43)                       Ree–Hoover virial expansion and hard particles
and that furthermore if we denote the surface area of the D-dimensional hypersphere by                       Ree–Hoover virial expansion and hard particles that
(7.44)                       Ree–Hoover virial expansion and hard particles

To evaluate CD we consider the integral

(7.45)                       Ree–Hoover virial expansion and hard particles

This integral is also expressed in terms of CD as using

(7.46)                       Ree–Hoover virial expansion and hard particles
as
(7.47)                       Ree–Hoover virial expansion and hard particles

Therefore

(7.48)                       Ree–Hoover virial expansion and hard particles
and thus we have
(7.49)                       Ree–Hoover virial expansion and hard particles
and
(7.50)                       Ree–Hoover virial expansion and hard particles

(p.191) Thus in any dimension D we have the desired answer

(7.51)                       Ree–Hoover virial expansion and hard particles

For large D we use Stirling's formula that for z → ∞

(7.52)                       Ree–Hoover virial expansion and hard particles
to obtain
(7.53)                       Ree–Hoover virial expansion and hard particles

For low dimensions B 2 is explicitly given in Table 7.5.

Table 7.5 Exact and decimal results for B 2 and B 3 for 2 ≤ D ≤ 12

                      Ree–Hoover virial expansion and hard particles

7.3.2 Evaluation of B 3

The third virial coefficient is given by

(7.54)                       Ree–Hoover virial expansion and hard particles

Since f(r) is −1 for                       Ree–Hoover virial expansion and hard particles and 0 for                       Ree–Hoover virial expansion and hard particles,

(7.55)                       Ree–Hoover virial expansion and hard particles

The above integrand either has the value 1 or 0, and r1 is now constrained to be within a ball of radius σ centered at the origin. Then r2 must be placed within distance a (p.192) from the origin, and also within unit distance from the point r1. The integral over all possible r2 thus corresponds to the volume of intersection of two D-dimensional spheres of radius a separated by a distance of                       Ree–Hoover virial expansion and hard particles. Rescaling all distances by σ, replacing the integration variable r 1/σ by u, and denoting the volume of intersection of the two hyperspheres of unit radius by V(u),

(7.56)                       Ree–Hoover virial expansion and hard particles
where ΩD is given by (7.50). To calculate V(u) integrate the one-dimensional overlap L(h) of two line segments with radius 1 at a separation of h, over the D − 1 dimensional hyperplane that is the perpendicular bisector of the line connecting the two hypersphere centers. This integration region is illustrated in Fig. 7.3

                      Ree–Hoover virial expansion and hard particles

Fig. 7.3 Integration over h for B 3.

We see from Fig. 7.3 that

(7.57)                       Ree–Hoover virial expansion and hard particles

Therefore

(7.58)                       Ree–Hoover virial expansion and hard particles
and hence (p.193)
(7.59)                       Ree–Hoover virial expansion and hard particles

Substitute sin φ = uh for h, and then reverse the order of integration.

(7.60)                       Ree–Hoover virial expansion and hard particles

Integrating directly the first and second terms, and integrating by parts the third and fourth terms gives:

(7.61)                       Ree–Hoover virial expansion and hard particles

Thus, using (7.51) we find the result of [10]:

(7.62)                       Ree–Hoover virial expansion and hard particles

To complete the evaluation of the integral in (7.62) we use the identities

(7.63)                       Ree–Hoover virial expansion and hard particles
(7.64)                       Ree–Hoover virial expansion and hard particles
to find
(7.65)                       Ree–Hoover virial expansion and hard particles
(7.66)                       Ree–Hoover virial expansion and hard particles

(p.194) Thus for D = 2N we use the duplication formula for the gamma function

(7.67)                       Ree–Hoover virial expansion and hard particles
and (7.65) in (7.62) to obtain
(7.68)                       Ree–Hoover virial expansion and hard particles
and for D = 2N + 1 we use (7.66) and (7.67) in (7.62) to find
(7.69)                       Ree–Hoover virial expansion and hard particles

For low dimensions                       Ree–Hoover virial expansion and hard particles is explicitly given in Table 7.5.

To obtain an expansion of                       Ree–Hoover virial expansion and hard particles for large D we expand the integrand in the integral of (7.62) about Φ = π/3 to find

(7.70)                       Ree–Hoover virial expansion and hard particles
and thus, using the expansion as z → ∞
(7.71)                       Ree–Hoover virial expansion and hard particles
we obtain from (7.62) that as D → ∞
(7.72)                       Ree–Hoover virial expansion and hard particles

7.3.3 Evaluation of B 4

The evaluation of B 4 in even dimensions up through D = 12 was carried out in [11] using the Ree–Hoover formalism. The integrals are of the form of overlapping sphere volumes and generalize to B 4 the computations of the previous subsection for B 3. All (p.195) integrals involved are elementary but their evaluation was sufficiently tedious that to obtain explicit results an algebraic computer program was used.

However, for odd dimensions the straightforward application of this method leads to elliptic integrals in intermediate steps. This is unfortunate because the final result of van Laar [6] for D = 3 makes no reference to elliptic integrals. For D = 3 the evaluation of B 4 was carried out by Nijber and van Hove [7] by means of the “two center” formalism which was invented by de Boer [39] in 1949. The evaluation of B 4 by means of this formalism has been extended by Lyberg [12] to odd dimensions up through D = 11. Once again only elementary integrals are encountered which, as for even dimensions, are sufficiently tedious that explicit evaluation was carried out using algebraic computer programs.

The final results for 2 ≤ D ≤ 12 are given in Table 7.6. The most striking feature of B 4 is that it is positive for 2 ≤ D ≤ 7 and is negative for 8 ≤ D. This feature was first seen in numerical computations by Ree and Hoover [13] in 1964.

Table 7.6 Exact and decimal results for B 4 for 2 ≤ D ≤ 12. The first five digits of the decimal result are shown.

                      Ree–Hoover virial expansion and hard particles

7.4 Monte–Carlo evaluations of B 5-B 10

The preceding results on the second, third and fourth virial coefficients exhaust all known cases where hard sphere virial coefficients have been exactly computed. For B 5 exact analytic evaluation has only been done on selected diagrams of the Mayer type [37, 38].

To obtain further information we must turn to Monte–Carlo evaluations of the integrals on the computer. These studies were initiated by Ree and Hoover [1] in 1964 where the basic method is explained. The limits of the maximum order k for which Bk can be computed depend both on the speed and storage capacity of the computer used and on the efficiency of the algorithms for generating the diagrams and evaluating the integrals.

(p.196) The chronology of numerical computations of virial coefficients is summarized in Table 7.7. The most extensive computations are those of Clisby and McCoy [17] which are given in Table 7.8.

Table 7.7 Chronology of numerical computation of the virial coefficients B 5B 10.

Date

Author (s)

Property

1964

Ree, Hoover [1]

B 5, B 6 for D = 2, 3

1967

Ree, Hoover [3]

B 7 for D = 2, 3

1993

van Rensburg [14]

B 8 for D = 2, 3

1999

Bishop, Masters, Clarke [15]

B 5, B 6 for D = 4, 5

2006

Clisby, McCoy [17]

Bk for 5 ≤ k ≤ 10 and 2 ≤ D ≤ 12

Table 7.8 Numerical results for                       Ree–Hoover virial expansion and hard particles for 5 ≤ k ≤ 10 taken from [17]. The underline indicates the position of the local minima and maxima. The error in the last digits is given in parenthesis.

                      Ree–Hoover virial expansion and hard particles

Further insight is obtained by the examination of the contribution of the individual diagrams. We list these in Tables 7.9 and 7.10 for B 5 and B 6 The results for the individual diagrams' contributions to B 7 in D = 2, 3 can be found in [3].

7.5 Hard sphere virial coefficients for k ≥ 11

For virial coefficients Bk with k ≥ 11 it has not yet been possible to evaluate all diagrams. We conclude our study here by examining, for k ≥ 11, two particular diagrams (p.197)

Table 7.9 The contributions of diagrams B 5[m, i] given in Table 7.1 to the fifth virial coefficient for hard disks and spheres. The values given are the product of the integral and the combinatoric factor Ck[m, i]. The error in the last digits is given in parenthesis.

                      Ree–Hoover virial expansion and hard particles

Table 7.10 The contributions of the diagrams B 6[m, i] given in Table 7.2 to the sixth virial coefficient for D = 2, 3, 5, 6, 7. The values given are the product of the integral and the combinatoric factor Ck[m, i]. The error in the last digits is given in parenthesis.

                      Ree–Hoover virial expansion and hard particles

which in a sense represent two extreme cases.

1) For D = 2, for both B 5 and B 6, the largest diagram (by almost a factor of 10) is the complete star B 5[0, 1] and B 6[0, 1] where each point is connected to every other point by a bond f. Because the bonds f vanish when                       Ree–Hoover virial expansion and hard particles the points of this diagram are as close together as possible. We call such a diagram “close packed.” The contribution of this close packed diagram to Bk is positive for for all k.

2) For large D and any k the largest diagram is the “ring diagram” where there are only k bonds of type f which join the k together in a ring where each particle is connected to only two other particles and all other bonds are of type                       Ree–Hoover virial expansion and hard particles. For B 5 and B 6 these diagrams are given in Tables 7.1 and 7.2 as B 5[5, 2] and B 6[6, 3], and their (p.198)

Table 7.11 The ratio of the ring to the star diagrams                       Ree–Hoover virial expansion and hard particles. The error in the last digits is given in parenthesis.

Order

D = 2

D = 3

D = 4

D = 5

4

−0.02986(16)

−0.09417(35)

−0.19636(79)

−0.34048(67)

5

0.02375(16)

0.11433(66)

0.3618(32)

0.9111(55)

6

−0.02296(37)

−0.1844(20)

−0.912(10)

−3.471(48)

7

0.02373(46)

0.3368(69)

2.915(56)

1.877(52) × 101

8

−0.02773(56)

−0.732(20)

−1.041(29) × 101

−1.000(28) × 102

9

0.03496(76)

1.673(47)

4.36(12) × 101

7.06(32) × 102

10

−0.0428(10)

−4.07(11)

−1.767(67) × 102

−5.14(78) × 103

11

0.0559(15)

1.030(29) × 101

8.63(83) × 102

12

−0.0759(21)

−2.598(73) × 101

−3.04(76) × 103

13

0.1024(29)

6.01(49) × 101

14

−0.1330(49)

−1.60(30) × 102

15

0.1819(98)

16

−0.237(20)

17

0.387(48)

18

−0.53(11)

values are given in Tables 7.9 and 7.10. In these diagrams the particles are as far apart as they can possibly be. We call such a diagram “loosely packed.” The contribution of these diagrams to B k has the sign (−1)k−1.

Some insight into the sign of Bk may be gained by studying the ratio of the ring to the star diagram as a function of k and D. This has been done numerically [16] and the results for the ratio are given in Table 7.11. There we see that for D = 2 the star dominates the ring for k as large as 18. This is some indication that for D = 2 the virial coefficient Bk can be expected to be positive for at least k ≤ 18. However, for D = 3 the ring is slightly larger than the star for k = 9 and at D = 4 the ring dominates for k = 8. The dominance of the ring over the star diagram seems to be a (weak) necessary condition for Bk to have a possible negative sign. Using this as a criterion it may be possible to see a negative sign for Bk in D = 3, 4 for k not much larger than 9 but there is no evidence that, for D = 2, Bk can be negative for k ≤ 18.

7.6 Radius of convergence and approximate equations of state

The most important property of the virial coefficients Bk is not their actual values for k less than some finite number but rather their asymptotic behavior as k → ∞ because it is this asymptotic behavior which determines the radius of convergence of the series. In the preceding sections we have examined two properties of the large k behavior of Bk of hard spheres in D dimensions: the identical vanishing of Ree–Hoover diagrams due to geometric reasons seen in Table 7.3 and the loose packed dominance seen in Table 7.11. This therefore suggests the existence of two criteria which need to be fulfilled in order for Bk to be in the asymptotic large k regime:

Criterion 1

The number of nonzero Ree-Hoover diagrams has approached its large k behavior.

Criterion 2

The loose packed diagrams with the number of                       Ree–Hoover virial expansion and hard particles bonds near their maximum value numerically dominate Bk.

(p.199) We see from Table 7.3 for k = 10 that criterion 1 is only completely fulfilled for D = 2 and is not fulfilled at all for D ≥ 5.

We see from Table 7.11 that for D = 3 and k ≥ 12 that criterion 2 is well satisfied and that as D increases the criterion is satisfied for smaller values of k. However, for D = 2 the criterion is not satisfied even for k as large as 18.

We thus conclude that there is no dimension in which both of these criteria are fully satisfied although for D = 3 and D = 4 it is possible that both criteria could hold for some moderate value of k of the order of 12 to 14. It is thus probable that the true radius of convergence cannot be determined from the virial coefficients up to B10 given in Tables 7.5, 7.6 and 7.8.

Nevertheless it is of interest to see what results if we attempt to study the radius of convergence by means of the ratio method, and accordingly the ratios                       Ree–Hoover virial expansion and hard particles are plotted in Figs. 7.47.5 where the ratios have been normalized to the closest packed densities given in Table 7.12 instead of to the second virial coefficient B 2. In this table we also give the lower bound [(1 + e)2B 2]−1 (6.34) on the density of the fluid phase and the fluid and solid ends of the first order phase transition which are seen in the computer experiments to be presented in chapter 8.

Table 7.12 Values for hard spheres of diameter σ in dimensions D = 2, … , 8 of B 2, the density ρcp of the known densest packed lattices from [40, table 1.2], the fluid ρf and solid ρs ends of the computer-determined first order transition and the lower bound ρLB = [(1 + e)2B 2]1− (6.34) on the density where melting can occur.

                      Ree–Hoover virial expansion and hard particles

In Fig. 7.4 we plot the ratios for D = 2, 3 and see that the ratios appear to smoothly extrapolate to a radius of convergence at a value of the density which is greater than the close packed density ρcp.

In Fig. 7.5 we plot the ratios for D = 4. Now the plots are not smooth but have oscillations which become increasingly strong as k increases. No useful extrapolation of these ratios is possible and it may be that negative virial coefficients will appear for higher k.

For D ≥ 5 there are oscillations of sign of the virial coefficients which, if this occurs in the true asymptotic behavior, indicates that the radius of convergence is not on the real axis.

There is no dimension for which a study of the ratio plots gives evidence that (p.200)

                      Ree–Hoover virial expansion and hard particles

Fig. 7.4 Ratio plot for hard sphere virial coefficients of Tables 7.5, 7.6, 7.8 in dimensions D = 2, 3.

                      Ree–Hoover virial expansion and hard particles

Fig. 7.5 Ratio plot for hard sphere virial coefficients of Tables 7.5, 7.6, 7.8 in dimension D = 4.

the asymptotic regime of large k has been reached and thus there is little point in attempting to use the first 10 terms in the virial expansion to study the analytic properties of the equation of state. Nevertheless there have been many attempts to fit the first eight terms of the virial expansion to an approximate equation of state. It is conventional to express these approximate equations of state in terms of the packing fraction
(7.73)                       Ree–Hoover virial expansion and hard particles

The packing fraction equals one at the density that would obtain if the hard spheres fill all space which is an unphysical density above ρcp.

(p.201) Some of the approximate equations of state for hard spheres have singularities at η = 1:

(7.74)                       Ree–Hoover virial expansion and hard particles
(7.75)                       Ree–Hoover virial expansion and hard particles
(7.76)                       Ree–Hoover virial expansion and hard particles
(7.77)                       Ree–Hoover virial expansion and hard particles
and some have multiple poles at ηcp
(7.78)                       Ree–Hoover virial expansion and hard particles
where the bn are given in Table 4 of [23].

Some have simple poles at complex values of η determined by Padé approximates determined from the eight virial coefficients of [14] such as

(7.79)                       Ree–Hoover virial expansion and hard particles
which has poles at
(7.80)                       Ree–Hoover virial expansion and hard particles

Some have branch point singularities at what is called the “random close packed density” ηrcp determined from a D-log Padé approximate such as [24]:

(7.81)                       Ree–Hoover virial expansion and hard particles
with ηrcp = 0.6435, or the form
(7.82)                       Ree–Hoover virial expansion and hard particles
where in [25] ηrcp = 0.6435, s = .84 and the cn obtained from [25, Table II] and where in [26] a wide range of values for ηrcp and s are obtained depending on which form of the D-log Padé is used.

It is clear from these many forms that the first eight virial coefficients are far from estimating either the true radius of convergence or the nature of the leading singularity of the virial series for hard spheres.

(p.202) 7.7 Parallel hard squares, parallel hard cubes and hard hexagons on a lattice

The derivation of the Mayer expansion of the virial coefficients presented in chapter 6 is valid not only for spherically symmetric potentials but also for potentials with a directional dependence as long as the orientation of the molecules is fixed and the integrations are carried out only over the center of mass coordinates. One such example is hard squares or cubes whose edges are all fixed to be parallel. The integrals for this problem are dramatically simpler than for hard spheres. The first seven virial coefficients were computed analytically in 1962 [27] (five years before B 7 was numerically evaluated for hard spheres [3]). We give the results in Table 7.13.

Table 7.13 Virial coefficients Bk for for parallel hard squares and cubes (with σ = 1) from [27].

                      Ree–Hoover virial expansion and hard particles

Table 7.14 Virial coefficients kBk for the hard square, simple cubic, fcc and bcc lattice gas with nearest neighbor exclusion from [28, 30].

k

kBk squares

kB k cubic

kBk fcc

kBk bcc

1

1

1

1

1

2

5

7

13

9

3

13

19

85

25

4

17

7

385

−87

5

−19

−149

1,261

−1,070

6

−175

−833

1,633

−3,910

7

−503

−3,569

−167,154

3,613

8

−695

−14,553

100,977

9

373

−53,405

308,041

10

633

−165,413

−1,828,761

11

−2,007

−4,400,021

−21,205,645

12

−58,207

13

−237,691

Furthermore the parallel hard squares (cubes) can be restricted to lie on a square (p.203) (cubical lattice). In this case the evaluation of the integrals reduces to a problem of pure combinatorics, and substantially more terms can be obtained.

The case of the lattice gas of hard squares was studied in [28] and in three dimensions the simple cubic, fcc and bcc lattice were studied in [30]. The results are given in Table 7.14. Note that for hard squares, terms up to order 42 can be obtained from the work of [29]. Even more can be computed for hard hexagons on the triangular lattice

Table 7.15 Virial coefficients Bk for the hard hexagon lattice gas from table 11 of [33] where kBk = −(k−1)βk−1.

k

kBk

1

1

2

7

3

31

4

115

5

391

6

1237

7

3529

8

8155

9

8311

10

−6, 2543

11

−612,809

12

−3,759,551

13

−19,472,387

14

−91,607,873

15

−402,535,529

16

−1,671,753,125

17

−6,585,730,265

18

−24, 544,637,087

19

−85,671,502,739

20

−273,505,952,615

21

−753,160,139,729

22

−1,456,884,883,535

23

860,351,408,035

24

30,699,547,973,425

25

288,155,349,143,341

which has the remarkable property discovered by Baxter [31, 32] that it can be solved exactly for all densities. This exact solution will be discussed in chapter 15 but it is useful here to note that the radius of convergence of the low density virial expansion is determined by a singularity in the complex ρ plane at
(7.83)                       Ree–Hoover virial expansion and hard particles

(p.204) The first 25 virial coefficients have been determined by Joyce [33] and are given in Table 7.15.

There are distinct differences in the patterns of signs of the Bk for the hard spheres of Table 7.8 and the hard squares, cubes and hexagons of Tables 7.137.15.

For the hard hexagons of Table 7.15 the signs oscillate with a rather large period and because the location of the leading singularity is known exactly (7.83) to lie in the complex plane near the real axis this oscillation will continue indefinitely as k → ∞.

For hard squares the signs of the Bk in Table 7.14 also oscillate and because of the similarity with the hard hexagons of Table 7.15 it is very natural to conjecture that this oscillation also contributes for all k and that the leading singularity is in the complex plane. The cubic and bcc lattice also have this property.

For the parallel hard cubes of Table 7.13 it is tempting to interpret the negative signs of B 6 and B 7 as the beginning of an oscillation that will give a leading singularity in the complex plane.

However, for the parallel hard square results in Table 7.13 and the hard sphere results in Tables 7.5 and 7.6 the interpretation is less clear. The signs of the parallel hard square virial coefficients and the hard spheres in D = 2, 3, 4 are always positive which is consistent with a leading singularity on positive real axis while the (−1)k−1 oscillation of sign for hard spheres with D ≥ 5, if carried out to infinity, would put the leading singularity on the negative real axis.

7.8 Convex nonspherical hard particles

We finally note that for potentials that are not radially symmetric the more physically relevant problem will integrate over the orientation of the molecules as well as their center of mass.

The first computation of this kind was done by Onsager in 1949 [34] who found that the second virial coefficient for a cylinder of diameter d and length l which is capped on both ends by a hemisphere of diameter d is

(7.84)                       Ree–Hoover virial expansion and hard particles

Boublik [35] showed in two dimensions that in general for hard bodies with proper area A and perimeter s the second virial coefficient is

(7.85)                       Ree–Hoover virial expansion and hard particles

The cases of ellipses with semi-major axes                       Ree–Hoover virial expansion and hard particles, rectangles with sides                       Ree–Hoover virial expansion and hard particles and                       Ree–Hoover virial expansion and hard particles and needles (the limiting case of rectangles where                       Ree–Hoover virial expansion and hard particles of length l have been studied in [36]. The second virial coefficients are (with A = 1 for ellipses and rectangles):

ellipses

(7.86)                       Ree–Hoover virial expansion and hard particles
where
(7.87)                       Ree–Hoover virial expansion and hard particles

(p.205) rectangles

(7.88)                       Ree–Hoover virial expansion and hard particles

needles

(7.89)                       Ree–Hoover virial expansion and hard particles

For needles the third virial coefficient has been analytically evaluated as [36]:

(7.90)                       Ree–Hoover virial expansion and hard particles

For ellipses and rectangles the virial coefficients B 3 and B 4 have been evaluated by Monte–Carlo [36] and are given in Table 7.16. However, it should be noted, that many, if not most, molecules do not have a convex shape and that there is no example of a nonconvex body for which even the second virial coefficient has been computed.

Table 7.16 Virial coefficients                       Ree–Hoover virial expansion and hard particles and                       Ree–Hoover virial expansion and hard particles for hard ellipses and rectangles as a function of the aspect ratio from [36].

                      Ree–Hoover virial expansion and hard particles

7.9 Open questions

We concluded the preceding chapters on order and scaling theory by presenting a selection of results which physical intuition suggests should hold but for which no proof has yet been found. We referred to these as “missing theorems.” The study of hard particles in this present chapter also has revealed a large number of places where desired results are missing. However, in contrast to the cases of order and scaling, here we have little or no intuition as to what results are to be expected and consequently we here discuss “open questions” in place of “missing theorems”.

The most important property of the virial series we would like to know is the location of the leading singularity and the radius of convergence. If the virial coefficients are all positive the leading singularity will be on the positive density axis. Unfortunately we found in section 7.6 that while the the virial coefficients for hard spheres in D = 2, 3 up through B 10 are all positive they extrapolate to give a radius of convergence which is greater than the close packed density. Thus it is plausible that Bk is not in an asymptotic large k regime for k = 10.

(p.206)

Table 7.17 Open questions for virial expansions of hard particles.

  1. 1. Are there negative Bk for hard discs and spheres?

  2. 2. What fraction of Ree–Hoover diagrams for Bk vanish identically for large k?

  3. 3. What is the analytic expression for B 4 for arbitrary D?

  4. 4. Can Bk be evaluated analytically for hard discs and spheres for k ≥ 5?

  5. 5. What is the true radius of convergence for hard spheres in D dimensions?

                      Ree–Hoover virial expansion and hard particles

Fig. 7.6 Ratio plot on an expanded scale for hard sphere virial coefficients in dimensions D = 3. The point at k = 7 is very slightly above the line joining k = 6 and k = 8.

Indeed, because negative Bk occur for hard spheres for D ≥ 5, for parallel hard cubes and for the lattice gases with nearest neighbor exclusion in both D = 2 and D = 3 it seems fair to suggest that negative virial coefficients generically occur for hard particles and that it will require some special property in order to make the coefficients all positive. However, we see in Table 7.15 that, for hard hexagons, the first negative coefficient occurred only at B 10 so that a large value of k may be necessary to see a negative Bk in D = 2. However, for D = 3 a close inspection of the ratio plot in Fig 7.6 shows that there is a slight deviation from monotonicity in the slopes and this may indicate the beginning of oscillations such as are seen in the ratios for D = 4 of Fig. 7.5 which can build up to eventually give changes in sign of Bk.

It is furthermore to be noted that it is not particularly satisfactory that for the analytic evaluation of B 4 we needed to resort to the use of algebraic computer programs. This prevents us from determining an analytic formula for arbitrary D.

Finally the question can be raised as to whether it is possible in principle to analytically evaluate all Bk for hard spheres in D dimensions in terms of simple numbers such as                       Ree–Hoover virial expansion and hard particles and π−1 arccos(1/3) as was the case for B 4. For B 5 seven of the ten Mayer diagrams given in appendix A of chapter 6 have been evaluated for D = 3 in [37] and [38] and are given in Table 7.18. Because hard sphere virial coefficients are geometrical objects related to the overlap of spheres it is appealing to conjecture (p.207)

Table 7.18 The known evaluations of the 10 Mayer diagrams (in the notation appendix A of chapter 6) which contribute to the fifth virial coefficient for hard spheres. The results for diagrams 1–5 are from [37]. The results for diagrams 6 and 7 are from [38].

                      Ree–Hoover virial expansion and hard particles

that such analytic evaluations are possible for all diagrams. In particular it should be remarked that a crucial step in the Hales proof of the fcc densest packing of hard spheres discussed in chapter 4 is the reduction of the problem to a finite number of configurations. It would be of great importance if this finitization has relevance to the computation of the virial coefficients.

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