Vol.3, No.7, 622-632 (2011) Natural Science
Copyright © 2011 SciRes. OPEN ACCESS
Closed virial equations for hard parallel cubes and
Leslie V. Woodcock
Instituto Cince Exacte, Departamento de Fisica, Universidade Federal de Juiz de Fora, Juiz de Fora, Brazil;
*Corresponding Author: les.woodcock@manchester.ac.uk
Received 20 May, 2011; revised 15 June, 2011; accepted 23 June, 2011.
A correlation between maxima in virial coeffi-
cients (Bn), and “kissing” numbers for hard hy-
per-spheres up to dimension D = 5, indicates a
virial equation and close-packing relationship.
Known virial coefficients up to B7, both for hard
parallel cubes and squares, indicate that the
limiting differences Bn – B(n-1) behave similar to
spheres and disks, in the respective expansions
relative to maximum close packing. In all cases,
the difference Bn – B(n-1) is approaching a nega-
tive constant with similar functional form in
each dimension. This observation enables
closed-virial equations-of-state for cubes and
squares to be obtained. In both the 3D and 2D
cases, the virial pressures begin to deviate from
MD thermodynamic pressures at densities well
below crystallization. These results consolidate
the general conclusion, from previous papers
on spheres and disks, that the Mayer cluster
expansion cannot represent the thermodynamic
fluid phases up to freezing as commonly as-
sumed in statistical theories.
Keyw ords: Virial Equations Liquid Theory
An equation-of-state for the hard-sphere fluid pressure
(p) was obtained by analytical closure of the virial ex-
pansion in powers of density relative to close-packing
[1]. The virial series is simply a Maclaurin expansion of
the pressure as a state function about zero density, and
can be written in dimensionless form
 
where Z = pV/NkBT, T is absolute temperature, kB is
Boltzmann’s constant, Bn are the dimensionless coeffi-
is the number density
NV , and 0
the crystal close packing density. The theory of the coef-
ficients Bn is well-established; exact statistical expres-
sions for Bn are available as the cluster integrals that are
calculated analytically up to B4 and presently available
numerically up to B10.
A closed form of Eq.1 based only upon known coeffi-
cients, B1 to B10, has been shown to be everywhere con-
vergent up to its first pole at 0
. The equation-of-state
is accurate for the equilibrium hard-sphere fluid, yield-
ing the same thermodynamic pressures as may now be
obtained with 6-figure precision from MD simulations of
up-to a million spheres [2]. A more refined scrutiny of
the margins of difference, with due consideration of the
uncertainties, however, shows that the virial equation
begins to deviate from the thermodynamic pressure
equation, albeit very slightly, at a density in the mid-
fluid range close to the available volume percolation
transition [3].
It has also been shown that the known virial series of
the two-dimensional (D = 2) hard-disk fluid is also
amenable to closure [4]; an equation-of-state is obtained
with 6-figure accuracy at low density. As in the D = 3
case, but more pronounced however, there is a clearer
deviation well-below the freezing transition. A subse-
quent more accurate analytical closed form for the pres-
sure equation-of-state for hard disks was derived [5].
The precision of this revised version of the closed virial
equation was such that we were able to conclude that the
deviation from the thermodynamic pressure occurs at or
near the 2D-percolation density of available, or excluded
volume, 2
pa 04.
, determined previously by Hoover
et al. [6]. Thus, for both disks (D = 2) and spheres (D =
3) the virial equations-of-state do not represent the
thermodynamic state functions for densities above the
respective free volume percolation transitions.
It is also worth noting that the known virial coeffi-
cients for D = 4 hyper-spheres [7], in powers of density
relative to close packing, also increase from B1 B6,
then decrease with Bn B(n-1) linearly with n, going
negative at n = 10. This indicates that the higher virial
coefficients should eventually go negative, yielding a
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
closed virial equation with a negative pressure pole at
maximum packing in this case also.
These empirical observations regarding the destiny of
the virial equations-of-state at higher densities would
appear to confirm that the percolation transitions in these
hard-core model fluids may indeed be higher-order
thermodynamic phase transitions that signify the onset
of the divergence of the virial pressure equation from the
thermodynamic pressure. There are those who seem to
“religiously” believe the Mayer virial series is equivalent
to the thermodynamic fluid equation-of-state; this is a
common misapprehension amongst physicists.
The suggestion that the virial expansion may not rep-
resent the physical thermodynamic state for densities
above the percolation transition is not new. For both hard
parallel cubes and hard parallel squares, the virial coeffi-
cients are known up to B7 [8-10] In fact, it has been
known for 50 years that the virial coefficients for parallel
cubes go negative [10]. MD calculations 25 years ago
[11], of the thermodynamic and transport properties of
hard parallel cubes, indicated that the hard-parallel-cube
fluid virial equation deviates above the percolation tran-
sition, at a density well-below the freezing transition.
Kratky [12] further elucidated upon the suggestion the
percolation transitions in spheres and disks may be
higher order thermodynamic phase transitions. Refer-
ence [12] also gives valuable definitions and discussion
of the various percolation transitions in hard-core fluids
On revisiting the known virial coefficients for both
fluid systems of parallel cubes and squares (Table 1), it
can be seen that in both cases, now in the expansion of
the packing fraction (y =
) even from just the
known coefficients up to B7, the differences Bn B(n-1)
are approaching negative constants with functional
forms analogous to the D = 3 spheres [1-3] and D = 2
disks [4,5] cases respectively. Here we obtain and test
the closed virial equations-of-state for the systems of
parallel cubes and squares, which by analogy with the
closed equations of spheres [1-3] and disks [4,5], and
then compare with available MD data from the literature.
The first simple observation we make, about the Bn
values in Table 1, is that for both D = 3 and D = 2, Bn
first increases with n, peaks at n = 4 (3D) or 5 (2D) and
then begin to decrease, and in the case of the cubes go
negative. This is just the same behavior that is seen gen-
erally in hyper-spheres of D > 1, i.e. D = 2, 3, 4 and 5.
The highest virial coefficient in all these cases is of
the same order as a maximum co-ordination or contact
numbers, known as “kissing numbers” to mathemati
Table 1. Known virial coefficients of the parallel hard-cube
fluid (D = 3) and parallel hard square fluid (D = 2): also given
in columns 3 and 5, for D = 3 and 2 respectively, are the pre-
dicted values up to B12 from the closed virial equations ob-
tained here.
n Bn [D = 3] Eq.3 Bn [D = 2] Eq.6
1 1 1
2 4 2
3 9 3
4 11.33333 3.666666
5 3.16 3.7222
6 –18.88 –18.88 3.025 3.0407
7 –43.5 –43.50 1.65 1.6714
8 –69.00 –0.3857
9 –94.50 –3.1306
10 –120.00 –6.5633
11 –145.50 –10.6838
12 –171.00 –15.4921
cians [12], of contiguous spheres to a central sphere.
This correlation provides salient evidence that the virial
expansions about zero density, in powers of density rela-
tive to close packing, Eq.1, reflects information about
the close packed state, and is per se suggestive of the
closed forms with the first poles at that maximum den-
The kissing number is simply defined as the maxi-
mum number of hyper-spheres in a configuration of like
spheres that may be contiguous with a central sphere at
the same time. This number is not necessarily the lattice
near-neighbor number, but for the lower dimensions up
to D = 5, it can represent the first co-ordination number
in the most stable crystal structure at infinite pressure, or
equivalently, at zero temperature. Kissing numbers for
hyper-spheres are known up to D = 28 [12] but here
(Figure 1) we compare with D = 1 to D = 5. The corre-
lation is convincing evidence that the virial expansion of
non-space-filling hyper-spheres does not reflect the
physically unreal packing (y = 1), seen in many theo-
retical approximate virial equations, but rather the
maximum crystal close-packed states (0
Looking again at Table 1, the Bn values for the cases
D = 3 (cubes) and D = 2 (squares) have similar maxima
at B4 (D = 3) value 11.3333 and B5 (D = 2) value 3.7222.
There are three different types of “kisses” in these sys-
tems, faces, edges and corners. Counting only faces and
edges, we get for Bmax + D, 14.3 and 5.7, compared to 14
and 4 for the respective kissing numbers of cubes and
squares respectively. In these cases, the density relative
to close packing
0 = packing fraction y, and the
maximum density corresponds to y = 1.
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
Kissing numbersof hard hyperspheres
dimension (D)
kissing numbe
kissing numbers
virial maximum
Figure 1. Kissing numbers for D-dimensional hyper-spheres
(shown as red circles centered on the exact whole number)
compared to the values of Bn(max) + D (shown as smaller
black circles); the maximum virial coefficients for D = 2 to 5
are 4.243 (B9), 8.864(B8), 22.639(B6) and 39.784(B5) respec-
Below, we will use y to represent reduced number
density, and anticipate a singularity in the closed virial
equations at y = 1.
The values of the known virial coefficients for hard
parallel cubes are listed in Table 1. Although presently
limited to B7, the values B5 to B7 are already indicative
of a similar closed functional form to that obtained for
hard-spheres [1,2]. In Figure 2, when plotted against
exp(–n), by analogy with spheres in reference [2], the
incremental values Bn – B(n-1) beyond (B5 – B4) decrease
assymptotically as n according to
 
01 2
BB 2
nn- A A expn A expn 
and approach a constant value (–25.5) which B7 – B6 ( =
–24.62) is almost upon. Eq.3 corresponds to the EXCEL
trendline fitted equation in Figure 2 below.
Neglecting the higher order term, once A0 is obtained,
the constant A1 is determined from Bm – Bm-1 and can be
used to predict all higher values Bn from m + 1 to infinity,
thereby effecting an analytic closure to the virial expan-
As with hard spheres, interpolation of the known
virial coefficients suggests that since the limiting value
of Bn Bn-1 is negative, the virial coefficients beyond B7
will remain negative. The corresponding virial equa-
tion-of- state will be continuous in all its derivatives,
eventually showing a negative pressure, with the first
pole at
0, i.e. y = 1.
Figure 2. Difference between successive virial coefficients (Bn
– Bn-1) from n = 7 to n = 5 for Bn in the expansion in powers of
the density relative to close packing: the difference Bn – Bn-1
decreases exponentially, and appears to asymptotically appro-
aches a constant value (–25.5) when n.
The virial equation-of-state for hard parallel cubes
would then take the form
 
1B B)
mn-m n-
yAA e y
 
 (4)
In which each term of the second summation can be
closed separately as with spheres, [2,3] to obtain an
equation-of-state. Eq.4 enables the closure for any
known n greater or equal to m = 7. The last term is a
negligible correction for finite m and disappears for
large m. It would appear from the data in Figure 1 and
Table 1 that m = 7 is sufficiently large for accuracy of
the same order as that of the available MD pressures in
this case, which is about 4-figure precision. Accordingly,
if we close the summation at m = 7 and put Ao = –25.5,
Eq.4 reduces to a form which is the same as the closed
virial equation for hard spheres [1,2] with A0 for cubes
fixed at –25.5 (Figure 2).
 
1B B
 
This equation-of-state for parallel cubes (Eq.5) with
A0 = –25.5) which can now be compared with the litera-
ture values of available thermodynamic pressures ob-
tained from MD simulations [11,14]. The virial equation
it accurate to within the uncertainty in the experimental
data up to the density around 0.2 to 0.3 and then it be-
gins to deviate. The pressure peaks, then decreases and
goes negative, and eventually diverges with a negative
pole at the density of maximum packing. The difference
in pressure between the closed virial Eq.5, with A0 =
–25.5, and the thermodynamic pressure from MD com-
putations in the vicinity of melting is shown in Figure 4.
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
Figure 3. Closed-virial equation-of-state (Eq.4) with m = 7
and A0 = –25.5: solid line) compared with thermodynamic
pressures (Z = pV/NkBT) obtained from MD simulations by
Woodcock and van Swol [11] (red circles) and Hoover et al.
[14] ( blue circles).
Figure 4. Deviation of closed-virial equation-of-state (Eq.5):
m = 7, A0 = 25.5) and thermodynamic pressures obtained from
MD simulations by Woodcock and van Swol [11] (red circles)
and Hoover et al. [13] (blue circles): DZ = ZMD – Z(virial).
Inspection of the discrepancy, between closed virial
Eq.5, and thermodynamic MD pressures, in Figure 4,
suggests that the virial pressure begins to deviate from
the thermodynamic pressure at a density on or below the
available volume percolation density (ypa) [11]. This
deviation is statistically significant; it would appear from
the forgoing analyses that it cannot be explained either
by uncertainties in the thermodynamic pressures, or er-
rors in the known virial coefficients, or any combination
of both. To do so the virial coefficients B8 and beyond
would have to take an extraordinary unusual twist. As
with spheres, however, we cannot say that within the
uncertainties, the deviation does not begin at an even
lower density, perhaps at the lower percolation transition
density (ype) associated with the excluded volume (see
also references [3,12]).
Turning now to the 2-D case of hard parallel squares,
inspection of incremental values of successive virial
coefficients in Table 1 shows that squares are behaving
similar to 2-D disks. Differences in successive Bn, in
powers of density relative to crystal close packing in
Eq.1, Bn Bn-1 are plotted in Figure 5. With only the
three points available the graph that beyond (B5 B4)
the increment decrease varies according to
nn- α αn
Eq.6 is the same functional form that was obtained for
hard disks where the virial coefficients are known up to
B10. As with disks, this interpolation of the known virial
coefficients suggest that since the limiting constant 0 is
negative, the virial coefficients will eventually become
negative and the corresponding virial equation-of- state
will eventually give a negative pressure, with the first
pole at y = 1, as with cubes. Eq.6 with the parameters
obtained from the EXCEL trendline in Figure 5 predicts
the first negative coefficient for squares will be B8. In
the corresponding closed-virial equation for D = 2 disks
B31 is predicted to be the first negative coefficient [5].
By analogy with disks, then the analytic closed form
virial equation-of-state for the hard parallel square fluid
takes the same form as that of the hard-disk fluid as fol-
lows (the derivation is given in the APPENDIX of ref-
erence [5]).
 
 
where Bm is the highest known virial coefficient, pres-
ently B7.
Using only the known coefficients B5 to B7 as given in
Table 1, the constants in the closed virial equation for
parallel squares, from the limiting value of Bn – B(n-1)
Figure 5, are
0 = –4.90, and the slope α = 24.9. The
closed virial equation-of-state for the pressure of the
parallel-square fluid can then be compared with the
thermodynamic pressure from the recent MD computations
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
Figure 5. Difference between successive coefficients (Bn – Bn-1)
from n = 7 to n = 5 in the expansion in powers of the density
relative to close packing: the difference Bn – Bn-1 is decreasing
roughly as 1/n, and would approach the constant –4.8973.
of Hoover et al. [14]. The virial pressure deviates from
the thermodynamic pressure at a density well-be- low
the freezing transition (around y = 0.7) eventually gives
an unrealistic negative pressure and diverges with a
negative pole at maximum close packed density 0 cor-
responding to y = 1.
In Figure 7, the deviation of Eq.7 for squares from
MD pressures [14] are plotted as a function of density
for all the MD data points above in Figure 3 except
the data point at the density 0.35 which is an aberration
that may contain an error. This plot suggests that the
deviation is originating in the vicinity of a low density
percolation transition. We have not seen a report of the
determination of this percolation transition for squares,
but the deviation is around the same value of
0 (= y
for squares) as that found by Hoover et al. to be the on-
set of the free volume percolation transition for the D =
2 hard-disk fluid [6].
In this paper we have looked at the trend in Bn Bn-1
for the known virial coefficients of cubes and squares in
the Mayer virial expansion Eq.1 and observed that the
same closed virial equations exhibit the same functional
forms, as has been obtained for spheres and disks re-
spectively [1-5].
When the resultant closed vrial equations are com-
pared with available thermodynamic pressures, as with
the fluids of spheres and disks, the virial pressure begins
deviating from the thermodynamic pressure at a low
fluid density. The percolation transition associated with
free volume has only been estimated for cubes, and has
not yet been reported for squares. Nonetheless, all the evi-
dence is that the onset of the deviations may be associ-
ated with higher-order thermodynamic phase transitions.
Figure 6. Closed-virial equation-of-state for a system of hard
parallel squares (Eq.4 with m = 7 and the parameters A and A0
as given in Figure 5: solid line) compared with thermodynamic
pressures (Z = pV/NkBT) obtained from MD simulations by
Hoover et al. [14] (blue circles).
Figure 7. Density dependence of pressure difference between
closed-virial equation-of-state (Eq.4: m = 10) and thermody-
namic pressures obtained from MD simulations by Hoover et
al. [14] for the system of hard parallel squares.
The APPENDIX to this paper illustrates the belief that
the virial expansion of Mayer [15] is actually equivalent
to the fluid equation-of-state of these hard core models,
at least up to freezing, is a widespread misapprehension
amongst theoretical physicists. In response to the various
suggestions that the empirical results of these closed-
virial comparisons are “speculative”, it seems that what
has been unduly “speculative” is the incorrect assump-
tion that Mayer’s cluster expansion represents the ther-
modynamic state functions of the fluid phases up to or
beyond the freezing transition. In the five cases we have
so far looked at, first spheres and disks [1-5], and now
here, squares and cubes, and also D = 4 hyper-spheres
(unpublished), the virial equations are deviating at a low
equilibrium-fluid density.
Many statistical theories are based upon the belief that,
if all the terms in the Mayer cluster expansion could be
approximated accurately, it would be a theory of “liq-
uids”. We now see that all these simple hard-core models
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
have two fluid phases, the low density gas phase where
the Mayer virial expansion represents the thermody-
namic state functions, and the high density fluid phase
where it is invalid. Looking again at the theory of simple
liquids, we may now conjecture that the high density
fluid phase belongs to the same phase as the supercooled
“liquid” phase, by definition. For hard-spheres this is the
metastable branch that is a continuous extrapolation of
the equilibrium high-density fluid at freezing, and which
terminates at the random close packed (RCP) state.
Perhaps we should now take another look at the RCP
state as a starting point for the general theory of liquids.
Standard treatises on simple liquids’ deal largely with
theories of the liquid state based upon “configurational
surgery” of Mayer virial cluster diagrams [15,16]. We
now see that the Mayer cluster expansion whilst being
an essentially exact theory of low density gases, may not
be a starting point for theories of the liquid state. An
analytical theory with all the virial coefficients correct
would not still represent the high density equilibrium
fluid. The title of Hansen and McDonald [16], when the
4th Edition comes to be published, might be retitled “The
Theory of Simple Gases”!
I wish to thank the Department of Physics at the Federal University
of Juiz de Fora for a temporary visiting facility, and my host Professors,
Socrates Dantas and Bernhard Lesche, for their support and kind hos-
[1] Woodcock, L.V. (2008) Virial equation-of-state for hard
spheres. ArXiv condensed matter.
[2] Bannerman M., Lue, L. and Woodcock, L.V. (2010)
Thermodynamic pressures for hard spheres and closed
virial equation-of-state. Journal of Chemical Physics,
132, 084507-084513.
[3] Woodcock, L.V. (2011) Percolation transitions in the
hard-sphere fluid. AICHE Journal (accepted and pub-
lished online). doi:10.1002/aic.12666
[4] Woodcock, L.V. (2008) Virial equation-of-state for the
hard disk fluid. ArXiv condensed matter.
[5] Beris, A. and Woodcock, L.V. (2010) Closed virial equa-
tion-of- state for the hard-disk fluid. ArXiv condensed
[6] Hoover, W.G., Hoover, N.E. and Hanson, Exact, K. (1979)
hard-disk free volumes. Journal of Chemical Physics, 70,
[7] Clisby, N. and McCoy, B.N. (2006) Ninth and tenth virial
coefficients for hard hyper-spheres in D-dimensions.
Physics and Astronomy, 122, 15-57.
[8] Zwanzig, R.W. (1956) Virial coefficients of parallel
square and parallel cube gases. Journal of Chemical
Physics, 24, 855-856. doi:10.1063/1.1742621
[9] Hoover, W.G. and De Rocco, A.G. (1961) Sixth virial
coefficients for gases of parallel hard lines, squares, and
cubes. Journal of Chemical Physics, 34, 1059-1060.
[10] Hoover, W.G. and De Rocco, A.G. (1962) Sixth and sev-
enth virial coefficients for the parallel hard cube model.
Journal of Chemical Physics, 36, 3141-3162.
[11] van Swol, F. and Woodcock, L.V. (1987) Percolation
transition in the parallel hard cube model fluid, Molecu-
lar Simulation, 1, 95-108.
[12] Kratky, K.W. (1988) Is the percolation transition of hard
spheres a thermodynamic phase transition? Journal of
Statistical Physics, 52, 1413-1421.
[13] Geometry: Multidimensional Geometry: n-Dimensional
[14] Hoover, W.G., Hoover, C. and Bannerman, M. (2009)
Single speed molecular dynamics of hard parallel squares
and cubes. Journal of Statistical Physics, 136, 715-732.
[15] Mayer, J.E. and Mayer, M.G. (1940) Statistical Mechan-
ics John Wiley, New York.
[16] Hansen, J.-P. and McDonald, I.R. (2006) The Theory of
Simple Liquids, 3rd Edition, Academic Press, Oxford.
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
APPENDIX 1. Debate on the Scientific Merit
of Reference [5]:
PRL anonymous Referee A said: “This manuscript
appears to me to be seriously flawed and either the flaws
must be corrected or it must be explained why my per-
ceived objections are invalid. My concerns are as fol-
1) It is rigorously established that the pressure is a
monotonic non decreasing function of the density.
Therefore it disturbing that the equation-of-state (4) di-
verges to negative infinity as 0
2) The authors will argue that their equation (4) is
called a “closed-virial equation-of-state'' to which the
mono-tonicity argument does not apply. This argument
itself is questionable and is in contradiction with every
approximate equation of state of which I am aware. If
the authors must persist in such a claim they must cite
the many papers which they are contradicting and ex-
plain why they are correct and the rest of the field is
3) The statement on page 5 that “the virial equation is
continuous in all its derivatives'' makes no sense. In par-
ticular if the system has a second or higher phase transi-
tion as the authors suggest then the virial expansion will
have a singularity (at which some derivative will diverge)
at the phase transition density; a density which will be
smaller than close packing.
4) On the purely numerical side of extrapolating virial
coefficients beyond the first 10 which have been com-
puted is it a completely unverified assumption that reli-
able extrapolations may be carried out to order 31 which
is where the authors claim that first negative coefficient
5) The authors seem to be completely unaware of the
long debated and studied question of the nature of the
transition in hard discs. At least they make no mention of
this literature in their bibliography. The authors may find
some of the references in their reference 4 relevant.”
Objection 1) is based upon a misapprehension. The
thermodynamic pressure of the hard-sphere fluid as de-
fined and derived from a partition function, and obtained
by MD simulations, is not necessarily the same as pres-
sure of a purely theoretical low-density-limit virial ex-
pansion. We do not know where the first phase transition
is in the thermodynamic equation-of-state of the
hard-disk fluid, but it must coincide with the initial de-
viation of the virial equation, beyond which the virial
equation must become physically unreal having no such
monotonic constraint.
Objection 2) is also invalid: there is no scientific
monatonicity argument for a virial expansioni in the
paper we point out that all the evidence is that the virial
coefficients of hard-core systems in higher dimensions,
including discs, go negative and can stay negative.
Objection 3) is again incorrect. The virial equation,
with the coefficients defined using the Mayer cluster
integrals, is an expansion about zero density which is
everywhere continuous in all its derivatives. The ther-
modynamic state function pressure, by contrast is not, it
will exhibit discontinuities at phase transitions.
Objection 4) appears to be prejudiced by the referee’s
misunderstanding above, and his failure to look at the
correlation data we present in Table 1 and Figure A1.
You only need to look at the numbers in Table 1 to see
the evidence for the closure is compelling, albeit em-
pirical. This behavior in two-dimensions is anticipated
by extrapolation from higher dimensions. The referee,
moreover, appears not to have seen Reference [1] (the
analogous science published in the JCP paper of the 3D
hard-sphere fluid).
Objection 5) could not be further from the truth! See
LAN LarXiv: 0806.1109 [pdf] 2008 Title: Two-phase
coexistence in the hard-disk model Authors: Leslie V
Woodcock Comments: Hard-disk controversy. In fact, I
have been interested in this phase transition for 30 years:
see Melting in two dimensions: determination of phase
transition boundaries” (Cape, van Swol and Woodcock)
Journal of Chemical Physics, Volume 73, 913-922 (l980).
Referee B said: “This paper presents some numerol-
ogy using the known virial coefficients to predict the
behavior of hard disks at higher densities than the con-
vergence of the known virial coefficients. Let me remind
the authors that non-analytical behavior is not predict-
able, such as phase transitions. Their results, though in-
teresting, are purely speculative, consequently. I would
suggest that the manuscript be considered for publication
in Physical Review E”
Please read the paper beyond the abstract?
Referee A again (9-point reply):
1) “The rebuttal letter of the authors emphasizes in
many places that the virial expansion will not agree with
the genuine thermodynamic equation-of-state beyond the
point of the first order fluid solid transition. This is, in
fact, exactly the reason which this present study is of
very little interest because all of the interesting behavior
which the authors report is indeed in the region where
the virial expansion no longer represents the physics of
the situation.
2) The authors rebuttal letter contains the following
phrase “PRL is not a review article; it would be inappro-
priate to cite the many previously proposed equations of
state of the hard disc fluid.” I find this sentence to be
extremely unprofessional. ALL scientific publications
have an absolute requirement to fully credit existing
work in the field. To say that a PRL is exempt from this
mandatory requirement is to diminish the scientific
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
credibility of the journal and must not be allowed. The
authors MUST demonstrate that their approach is correct
and the previous work is wrong.
3) The response to referee B is also very peculiar
where the author say that “a full PRE paper at this stage
is inappropriate.” It is my understanding the articles in
PRL are NOT supposed to be preliminary reports based
on partial or incomplete analysis but instead are intended
to be reports of completed research of exceptional inter-
est. The authors' response clearly demonstrates that the
research does not meet the criteria of being fully com-
4) I do agree with the criticism made by referee B that
the results are purely speculative. In my opinion purely
speculative articles should not be published in PRL.
5) The authors claim in their rebuttal letter that “all
evidence is that the virial coefficients of hard core sys-
tems in higher dimensions go negative and can stay
negative.” In the paper they say that “virial coefficients
for all hard hyper-spheres of higher dimensionality than
one eventually go negative.” This latter statement is not
true. In all computed cases the virial coefficients of odd
order are all positive.
6) In 2 dimensions there is a long standing contro-
versy as to whether or not there is a first or a second
order phase. This historical fact should be given refer-
ence. Indeed, there is no proof that a crystalline phase
exists at all (although there is also no proof that a crys-
talline phase is impossible either).
7) The authors assume without giving any reason at
all that the leading singularity in the virial expansion is
at the closest packing density and their conclusion that
the virial coefficients become and stay negative is driven
by this assumption. From the analysis of the Percus
Yevick equation it is quite likely that in odd dimensions
higher than the leading singularity is on the NEGATIVE
density axis which is consistent with the alternating
signs of the virial coefficients in dimensions 5 and
higher. Since a leading singularity at negative density
has nothing to do with close packing this referee does
not see why for dimensions 2 and 3 the leading singular-
ity can be assumed to be at close packing.
8) In the opinion of this referee this entire paper is
built on the assumption that the leading singularity in 2
dimensions is at close packing. Because this is a pure
assumption the second referee is completely correct in
pointing out that this paper is very speculative.
9) It is disingenuous of the authors not to give refer-
ence to the many papers which do not make this as-
sumption and predict virial coefficients which are all
positive. The authors are in effect claiming that most of
the literature on approximate equations of state is wrong.
I believe that if this were made clear by the authors that
the editors of PRL would agree that unless the authors
can clearly explain why they are right and where so
many others are wrong that the paper does not meet the
standards of the journal.”
Point 1 is a U-turn; Referee A evidently conceding his
original misapprehension, now turns towards political,
i.e. non scientific, reasons for rejection.
Points 2 and 3 are new, and non-scientific.
Point 4: the Referee A again misconstrues an empiri-
cal observation as some theoretical assumption”! There
is no theoretical assumption”, as there is, for example,
in the Percus Yevick theory and similar theories of sim-
ple liquids.
Point 5: the statement that virial coefficients can,
and in some will, go negative and stay negativeis cor-
rect and verifiable.
Point 6: scientists seeking to publish discoveries
should not be bound as a condition of publication to
make reference to irrelevant and often unprincipled,
scientific theory or scientific scams. There is now wide
acceptance by most principled scientists in the field of
statistical thermodynamics of condensed matter that the
work that this referee A keeps insisting we refer to, was
at best, a pie-in-the-sky theory, at worst a scam.
Point 8: there is nothing speculative about 1) an
empirical observation [Table 1 Figure A1] 2) an exact
mathematical derivation [Equation 4, APPENDIX 1],
and 3) a comparison with “experiment” (MD) [
A2 and A3.] In fact, there is no speculation in this paper
at all.
Point 9: it is nothing to do with standards”; it is
about discovering, albeit by empirical means, the correct
equation of state that reproduces the thermodynamic
pressure with the 6-figure accuracy that it can now be
computed up to the first thermodynamic phase transi-
Referee B again (reply): My review stands. The other
reviewer basically has the same objections as I and his
reply saying that his remarks only apply to the theoreti-
cal virial expansion does not make the paper physically
meaningful, though still of some interest.
B still hasn’t read the paper!
(new) Referee C said: “I basically agree with all the
comments of referee A and do not find the author's re-
sponse persuasive at all. I do not recommend publication
of this paper.”
Comment 2. “I am particularly concerned by points
(original objections) 3 and 4 raised by Referee A. Indeed,
in objection 3, referee A correctly points out that if there
is a phase transition at some density, then the virial ex-
pansion should have a singularity exactly at this density.
This is well known in statistical physics, for instance in
the study of the high temperature or high field expansion
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
of the 2-dimensional Ising model. On the contrary, in
this paper, the authors suggest that there might be a
phase transition while at the same time the low-density
expansion should stay convergent. This is impossible”
Comment 3: “In my opinion, the discrepancy between
MD data and the equation of state proposed by the au-
thors is only due to an inaccuracy of the extrapolation”
And indeed (point 4 of referee A) the extrapolation of
complex virial coefficients based only on the first 10
seems very dangerous. There are plenty of examples in
the mathematical literature where such an extrapolation
leads to completely wrong results”
Comment 2 is basically agreeing with the original
misapprehension of referee A, [Note: referee C doesn’t
know that referee A has since changed his mind] Not
only is it not impossible”, but that is exactly what does
happen. Moreover, this paper has nothing to do with
phase transitions of lattice gases that do not have the
same analytic virial expansion at zero density.
Comment 3 is meaningless without a relevant exam-
(new) Referee D said: “The paper presents an equa-
tion-of-state for the hard-disk liquid based on the first 10
virial coefficients and then notes that this equation devi-
ates from the simulated area-pressure curve at a density
of ~ 0.4, a value similar to that associated with the per-
colation of free volume. I cannot recommend publication
of the paper in PRL. My reasons are as follows:
1) The authors do not provide an explanation of the
deviation they report. The connection with the Ree -
Hoover results are an interesting speculation.
2) The authors fail to make any case for the signifi-
cance of their work. Even if one went beyond their paper
and regarded the connection between their observed de-
viation and the Ree-Hoover percolation as having been
established, why would this be important? I do not mean
to imply that it would not be important, merely that the
authors seem to have taken it for granted that the sig-
nificance is obvious.”
Point 1 overlooks the explanation based upon a
change in density fluctuation
Point 2 is also incorrect; there is no speculation”; it
is an empirical result! There’s always a chance that
could be an unfortunate coincidence but when you look
at the science, it’s highly unlikely.
Referee A again said, in response to a request to sup-
ply an example of the references he repeatedly insists
must be cited)
“The theory of 2 dimensional melting has a large lit-
erature not referred to by the authors. Some relevant
papers are 1) Kosterlitz and Thouless, J. Phys. C6 (1973)
1181-1203; 2) Halperin and Nelson, Phys. Rev. Lett. 41
(1978) 121-124; 3) Nelson and Halperin, Phys. Rev. B
19 2457-2484; 4) Binder, Sangupta and Nielaba, J. Phys.
Cond. Matt. 14 (2002) 2323-2333. To write on hard
discs without making contact with this long standing
problem is not acceptable. The authors have assumed
that the virial expansion has its leading singularity at the
close packed density. If this were correct then there is no
Kosterlitz Thouless transition and no hexatic phase. The
authors have not presented evidence to support this”
There is no longstanding problem”. In fact, the paper
has nothing to do with 2-dimensional melting, let alone
the fictitious hexatic mesophase to which referee A
refers. Papers 1) to 3) above do not withstand scrutiny:
the approximations in paper 1) and several of the equa-
tions in papers 2) and 3) cannot be validated. Papers 3)
above are effectively a scientific fraud that unfortunately
caught the imagination of a lot of “band-wagoners who,
over 30 years since, have published a lot of papers that
tell us nothing, see e.g. paper 4) above.
“Furthermore the conclusion is based on assumed
form for the large n behavior of the virial coefficients Bn
which leads to the conclusion for large n the virial coef-
ficients Bn will all be negative. This is a very strong
conclusion which 1) is totally dependent on their as-
sumed form and 2) is at variance with every other as-
sumed form of the virial coefficients in the literature. In
my opinion the authors do not give sufficient evidence to
support this conclusion.”
The form of Bn Bn-1 as a function of n, is not as-
sumed”, as the data in columns 3 - 4 of Table 1 confirm,
it is an empirical result.
Arbitration Report by PRL Divisional
Associate Editor: J. Machta (Nov. 2010)
“The authors have proposed a closed form equation of
state for 2D hard disks based on a fit to the known virial
coefficients. This paper provides an accurate equation of
state for hard disks that will be useful to workers in the
field. It also gives evidence for two interesting conjec-
tures: 1) the virial coefficients eventually become nega-
tive and 2) there is a free volume percolation transition
with thermodynamic consequences as proposed some
time ago by Hoover. Although this paper is interesting
and should be published, I do not recommend publica-
tion in PRL. The manuscript has been reviewed by four
independent, expert referees in the field. All of the refe-
rees definitively recommend against publication in PRL.
I am in agreement with the referees that the paper does
not meet the standards for publication in PRL.
1) A fitted equation of state by itself is not of suffi-
ciently broad interest to warrant publication in PRL.
2) I agree with the referees that the breakdown of the
virial equation of state and its relation to Hoover’s pro-
posal is too speculative for publication in PRL. To be
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
more specific, the deviations between the MD results
and the equation of state shown in Figure A3 are quite
small, always less than 10–3. On the other hand, fitting
over various rangesof n or adding a term of the form
b/n^2 in Eq.3 suggests to me that the error in the coeffi-
cients alpha and alpha_0 is in not smaller than 0.01 and
the accuracy is certainly not the 4 or 5 significant digits
quoted in the paper without error bars. If uncertainty of
this magnitude is put in Eq.4, it is sufficient to create
deviations qualitatively like those shown in Figure A3
without the need to invoke a breakdown in the virial
equation of state. There may also be systematic devia-
tions from Eq.3 that appear for virial coefficients be-
yond 10. Thus, I must consider the reality of a phase
transition at p = 0.4 to be speculative and not yet firmly
supported by the data at the level required for publica-
tion in PRL.
3. The authors clearly reject the idea of a continuous
transition to a hexatic phase. Nonetheless, this idea has
considerable currency in the community. The absence of
any mention of competing ideas makes the paper un-
suitable for a general readership.”
Reason 1: It is not a fitted equation-of-state. The
closed virial equation is derived from the known virial
coefficients without any fitted parameters .
Reason 2: Changing a or a0 by 0.01 or up to 2%
makes absolutely no difference to Figure A3. On an
EXCEL spreadsheet it is easy to test; below is an addi-
tion to the revised version.
This is Figure A3 with error bars.
Quote: “The deviations shown in Figure A3 are rather
insensitive to uncertainties of the order 1% in a and a0
The reason is that the contribution to Z from all un-
known virial coefficients (B11 - B) in the density range
from 0.4 to 0.7 is extremely small. DZ(B11 - B)/Z(total)
varies from 0.0003 at rs3 = 0.4 to 0.0477 at rs3 = 0.7.
Nevertheless, in order to be sure that the deviation be-
ginning at this low density is real, and not an artifact of
any uncertainties in the determination of the constants a
and a0 from the virial coefficients we have computed a
range of combinations of a and a0 for the full width of
uncertainty permitted by maxima and minima in all
known Bn respectively. The range of a and a0 values also
varies with the range of n used to obtain them. The low-
est and highest possible values obtained using only
known values B5 to B10 are 0.432414 :4.29360 and
0.441912:4.393061 for a:a0 respectively. The mean
values (0.43710:4.344 68) are very close to the best
EXCEL trendline for B5 to B14 shown in Figure A1. Thus,
the commencement of the deviation as shown in Figure
A3 cannot be caused either by errors in the known virial
coefficients, errors in the extrapolated contribution to Z
00.2 0.4 0.60.8
density (
Z(M D)-Z(v irial)
MD range of (0.7)
( 0.4)
density (
a =4.3919
Figure A1. Deviations for Machta values I.
00.20.4 0.6 0.8
density (
Z(M D)-Z(virial)
MD range of (0.7)
( 0.4)
density (
a =4.2181
Figure A2. Deviations from Machta values II.
from (B11 – B) or from uncertainties in the MD pres-
sures. We have to conclude that the deviation is not an
artifact of any of the data used to obtain the deviation
plot in Figure A3.”
End of QUOTE
Reason 3: There is no such thing as a continuous
transition to another phase in classical thermodynam-
ics; the above contradiction by Machta is as nonsensical
as the concept itself.
In summary, ALL of the three points above that DAE
Professor Machta cites as his reason for rejection are
{Machta was invited to withdraw reason 2 in the light
of the above evidence. In response he requested to see
Figure A3 redrawn with 4 different values of a and a0 he
These figures are reproduced below.
L. V. Woodcock / Natural Science 3 (2011) 622-632
Copyright © 2011 SciRes. OPEN ACCESS
00.20.4 0.6 0.8
density (
Z( MD)-Z ( virial)
MD range of (0.7)
( 0.4)
density (
a =4.3414
Figure A3. Density dependence of pressure difference between
closed-virial equation-of-state (Eq.4: m = 10) and thermody-
namic pressures obtained from MD simulations by Kolafa and
Rottner (Ref.2).The percolation transition density determined
by Hoover et al is indicated by the vertical arrow. The maxi-
mum uncertainties in the MD data for the thermodynamic
pressures as quoted by Kolafa and Rottner are also indicated as
horizontal lines; and range from ±0.000005 at the density 0.4
to ±0.000019 at the density 0.7. The solid and dashed lines are
continuous deviation error limits using the parametrized equa-
tion of state data using (EoS rmax = 0.88 ref.2) and calculated
from values of a and a0 obtained from the extremities of un-
DAE Machta request: Figure A3 be redrawn with
various different combinations of a and ao which he pre-
scribed; one combination which are here referred to as
the Machta parameters which he had evidently fitted to
remove the deviation between virial equation and the
Figure A4. Demonstration that the Machta-contrived values of
a and a0 predict incorrect Bn B(n1) values (upper line), com-
pared to the B-W equation (reference 5) and the known virial
coefficients (lower line).
thermodynamic pressures in an attempt to justify his
rejection reason 2 above.
There is no basis at all for a quadratic form, but it
seems almost certain that if it were used for all Bn > B11
it would give just the same result as seen here in Figures
Machta rebuttal Figure A3. Deviations for Machta
values IV
Arbitrarily adding 0.01 to a AND subtracting 0.1 (to
try to change the results to what Machta would like to
see) is improper. It is clear from the very small uncer-
tainty in B8 B
9 and B10 that the line must cross zero at
B10 - B9 which is 0 ± 0.0003 i.e. within a very small error
of uncertainty. Thus the proposed Machta combination is
well outside the uncertainty permitted by theerrors in the
known virial coefficients. This is clearly seen in the
EXCEL table and Figure 4.
Bn – Bn-1 C&M Bn Bn-1 (B-W) MACHTA
0.43424182 0.43441440 0.46442000 constant A0 constantA – A/A0
0.29335679 0.28971667 0.31638333
0.18676531 0.18636114 0.21064286 –0.432414 4.296300 9.9356127
0.10722915 0.10884450 0.13133750 –0.436730 4.363133 9.9904586
0.04530845 0.04855378 0.06965556 –0.440427 4.384234 9.9545078
–0.00319701 0.00032120 0.02031000 –0.441613 4.391239 9.9436362
–0.04279730 –0.03914182 –0.02006364 –0.441912 4.393061 9.9410312
–0.07267696 –0.07202767 –0.05370833 –0.439958 4.380815 9.9573482
–0.09235146 –0.09985415 –0.08217692 –0.434507 4.345766 10.001602
–0.12240047 –0.12370543 –0.10657857 –0.433772 4.340932 10.007405
MACHTA––––> –0.423800 4.441100 10.4792