Analytic methods for the Percus-Yevick hard sphere correlation functions
The Percus-Yevick theory for hard spheres provides simple accurate expressions for the correlation functions that have proven exceptionally useful. A summary of the author's lecture notes concerning three methods of obtaining these functions are presented. These notes are original only in par...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1199702017-06-11T03:04:34Z Analytic methods for the Percus-Yevick hard sphere correlation functions Henderson, D. The Percus-Yevick theory for hard spheres provides simple accurate expressions for the correlation functions that have proven exceptionally useful. A summary of the author's lecture notes concerning three methods of obtaining these functions are presented. These notes are original only in part. However, they contain some helpful steps and simplifications. The purpose of this paper is to make these notes more widely available. Теорiя Перкуса-Євiка для твердих сфер дає можливiсть отримати дуже кориснi простi точнi вирази для кореляцiйних функцiй. Тут представлено пiдсумковi нотатки лекцiй, що стосуються трьох методiв отримання цих функцiй. Представленi нотатки є тiльки частково оригiнальними. Проте, вони мiстять кориснi кроки i спрощення. Метою цiєї статтi є зробити цi нотатки бiльш доступними широкому загалу. 2009 Article Analytic methods for the Percus-Yevick hard sphere correlation functions / D. Henderson // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 127-135. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 02.30.Qy, 02.30.Rz, 05.20.Jj, 05.70.Ce, 64.30.+t DOI:10.5488/CMP.12.2.127 http://dspace.nbuv.gov.ua/handle/123456789/119970 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The Percus-Yevick theory for hard spheres provides simple accurate expressions for the correlation functions
that have proven exceptionally useful. A summary of the author's lecture notes concerning three methods of
obtaining these functions are presented. These notes are original only in part. However, they contain some
helpful steps and simplifications. The purpose of this paper is to make these notes more widely available. |
| format |
Article |
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Henderson, D. |
| spellingShingle |
Henderson, D. Analytic methods for the Percus-Yevick hard sphere correlation functions Condensed Matter Physics |
| author_facet |
Henderson, D. |
| author_sort |
Henderson, D. |
| title |
Analytic methods for the Percus-Yevick hard sphere correlation functions |
| title_short |
Analytic methods for the Percus-Yevick hard sphere correlation functions |
| title_full |
Analytic methods for the Percus-Yevick hard sphere correlation functions |
| title_fullStr |
Analytic methods for the Percus-Yevick hard sphere correlation functions |
| title_full_unstemmed |
Analytic methods for the Percus-Yevick hard sphere correlation functions |
| title_sort |
analytic methods for the percus-yevick hard sphere correlation functions |
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Інститут фізики конденсованих систем НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/119970 |
| citation_txt |
Analytic methods for the Percus-Yevick hard sphere correlation functions / D. Henderson // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 127-135. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hendersond analyticmethodsforthepercusyevickhardspherecorrelationfunctions |
| first_indexed |
2025-07-08T16:59:58Z |
| last_indexed |
2025-07-08T16:59:58Z |
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1837098863880568832 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 127–135
Analytic methods for the Percus-Yevick hard sphere
correlation functions
D.Henderson∗
Department of Chemistry and Biochemistry, Brigham Young University, Provo UT 84602
Received May 18, 2009, in final form June 9, 2009
The Percus-Yevick theory for hard spheres provides simple accurate expressions for the correlation functions
that have proven exceptionally useful. A summary of the author’s lecture notes concerning three methods of
obtaining these functions are presented. These notes are original only in part. However, they contain some
helpful steps and simplifications. The purpose of this paper is to make these notes more widely available.
Key words: Percus-Yevick, hard spheres, correlation functions, integral transforms, analytic results
PACS: 02.30.Qy, 02.30.Rz, 05.20.Jj, 05.70.Ce, 64.30.+t
1. Introduction
The hard sphere (HS) fluid is a simple representative fluid that, through perturbation theory [1,
2], is the basis of a simple and successsful theory of liquids. Hard spheres interact through the HS
potential
u(R) =
{
∞, R < d,
0, R > d,
(1)
where R is the separation of a pair of spheres and d is their diameter. The reason why the HS
fluid is so useful is that accurate analytic expressions can be obtained from the Percus-Yevick (PY)
theory [3]. Indeed, perturbation theory could have been developed and used decades earlier had the
PY expressions been available. The PY theory and the mean spherical approximation (MSA) [4]
are identical for the HS fluid. Since the MSA is more widely used, perhaps the PY results for hard
spheres should be called the MSA results. However, the name PY is conventionally associated with
these results so the conventional nomenclature will be used.
The PY theory produces analytic results for the HS thermodynamics, direct correlation function
(DCF), c(R), and the Laplace transform of the radial distribution function (RDF), g(R). The DCF
and RDF are related through the Ornstein-Zernike (OZ) equation,
h(R12) = c(R12) + ρ
∫
h(R13)c(R23)dr3 , (2)
where Rij = |ri − rj | is the separation of a pair of spheres located at ri and rj and ρ = N/V is
the number of spheres divided by the volume. The function, h(R) = g(R) − 1 is called the total
correlation function (TCF). Because of the discontinuity in the HS potential, it is useful to define
the background function or cavity function, y(R), by g(R) = e(R)y(R), where e(R) = exp[−βu(R)]
and β = 1/kBT with kB and T being the Boltzmann constant and temperature, respectively. The
background function is useful because it is continuous since the discontinuities of g(R) and c(R)
arise solely from e(R), which is zero for R < d and one for R > d. The integral in the OZ equation
is a convolution integral.
The OZ equation is a definition of the DCF and does not, by itself, yield a theory. The PY
theory for the HS fluid is obtained by combining the OZ equation together with both the exact
∗E-mail: doug@chem.byu.edu
c© D.Henderson 127
D.Henderson
result for the HS fluid,
h(R) = −1, R < d, (3)
and the approximation,
c(R) = 0, R > d. (4)
The DCF is, in a sense, an inverse of the TCF. To see this, the Fourier transform, f̃(k), of a
spherically symmetric function, f(R), is defined by
kf̃(k) = 4π
∫
∞
0
R sin(kR)f(R)dR (5)
with the inverse transform given by
Rf(R) =
1
2π2
∫
∞
0
k sin(kR)f̃(k)dk. (6)
Note that f̃(k) = f̃(−k). Making use of the fact that the Fourier transform of a convolution
integral is the product of the Fourier transforms of the constituent functions, the OZ equation,
when transformed, can be written as
1 + ρh̃(k) = [1 − ρc̃(k)]−1. (7)
The thermodynamic functions can be calculated from the correlation functions using the equa-
tions,
pV
NkBT
= 1 +
2π
3
ρd3y(d) (8)
and
1
kBT
∂p
∂ρ
= 1 − 4πρ
∫
∞
0
R2c(R)dR, (9)
where p is the pressure. Because c(R) and h(R) are related through the OZ equation, equation (9)
can also be written as
kBT
∂ρ
∂p
= 1 + 4πρ
∫
∞
0
R2h(R)dR. (10)
It is to be noted that the integrals in equations (9) and (10) are c̃(0) and h̃(0), respectively. Equation
(8) is exact for a HS fluid; equations (9) and (10) are exact for any simple liquid. Of course, in the
PY theory for a HS fluid, the upper limit for the integration in equation (9) is actually d. Equation
(8) is often called the pressure (p) equation or virial equation. Equation (9) [and equation (10)]
is often called the compressiblity (c) equation. Note that, for the HS fluid, y(d) = g(d+) is valid,
exactly. In the PY theory for hard spheres, c(d−) = −y(d). More generally, c(R) = −y(R), for
R < d, in the PY theory for hard spheres.
Another equation that is useful is
y(0) = −c(0) = 1 − 4πρ
∫ d
0
R2c(R)dR = β
(
∂p
∂ρ
)
(c)
. (11)
This equation, which we may call the PY zero separation equation, is valid only for the PY theory
for hard spheres. It is obtained from the OZ equation, equation (2), in the limit R12 = 0, together
with equations (3) and (4). The first term after the left equal sign is just h(0) = −1. The following
term is the integral in the OZ equation(in the limit R12 = 0), using h(R)c(R) = −c(R) for
R < d and zero for R > d. It is interesting to note that there is an exact zero separation theorem,
y(0) = exp(β∆µ), where ∆µ is the chemical potential of the hard sphere fluid in excess of that of an
ideal gas. This exact theorem follows from the fact that for hard spheres at zero separation, one hard
sphere has disappeared inside the other hard sphere and, in fact, one hard sphere has been removed
from the system. The exact theorm follows because the chemical potential is the energy needed
or gained to add or remove a particle from the system. The PY/HS theory does not even come
128
Analytic methods for the Percus-Yevick hard sphere correlation functions
close to satisfying the exact zero separation theorem. The combination of equation (11) and the
exact theorem would give a terrible result for the chemical potential. None-the-less, equation (11)
is useful for obtaining an expression for the PY/HS c(R).
Of course, if exact results were available, the (p) and (c) equations would be consistent. However,
because the PY theory is approximate, the pressure that results from the (p) and (c) equations will
be inconsistent and slightly different. The thermodynamic functions used in the zero separation
equation (11) are identical to the (c) equation results.
As has been stated, the PY theory is important for the HS fluid because it yields analytic
expressions for c(R) and explicit results for g(R) and y(R) that are quite accurate. Three methods
for obtaining these functions will be considered. The results presented here come from the author’s
lecture notes. One of the methods (the first) given here is probably original. However, when the
methods are not original, the steps and simplifications that are given have proven useful. For
example, some of these notes were found to be valuable by Gray and Gubbins and appear in an
appendix in their well-known monograph [5]. The author has decided to publish these lecture notes,
as a review, so that they might be more widely available.
2. Simple method
Even before the correlation functions for hard spheres were obtained, it was realized that the
virial coefficients for this fluid were particularily simple. The virial series is a power series in ρ for
the pressure,
pV
NkBT
= 1 +
∞
∑
n=2
Bnρn−1. (12)
The coefficients, Bn, are called the virial coeffients. Using the PY theory, Hutchinson and Rush-
brooke [6] obtained first seven Bn, as given by the p and c equations. Their results are given by
the formulae,
Bp
n =
2(3n− 4)
4n−1
bn−1 (13)
and
Bc
n =
1 + 3
2n(n − 1)
4n−1
bn−1, (14)
where b = 2πd3/3. The p equation virial coefficients are somewhat too small but are reasonably
good. The c equation virial coefficients are even more accurate. They are slightly too large.
Assuming these expressions to be generally valid, the power series can be summed to yield
(
pV
NkBT
)
(p)
=
1 + 2η + 3η2
(1 − η)2
(15)
and
(
pV
NkBT
)
(c)
=
1 + η + η2
(1 − η)3
, (16)
where η = πρd3/6 is the volume occupied by the spheres divided by the volume available and is
called the volume fraction of the fluid. It is quite possible that equations (13) and (14) could be
shown by induction to be correct to all orders of n. This has never been done. However, the other
methods considered in subsequent sections show that equations (15) and (16) are valid so there is
not much incentive for an induction proof. We will accept the above results in this section.
As has been observed for the virial coefficients, the (c) equation result for pV/NkBT is slightly
too large whereas the (p) equation result for this quantity is a little more in error and is somewhat
small. This suggests that a linear combination of the (p) and (c) equation results will give a good
description for the hard sphere pressure. Indeed, the well-known Carnahan-Starling formula, that is
given by pV/NkBT = (pV/NkBT )(p)/3+2(pV/NkBT )(c)/3, gives good agreement with simulation
results for the hard sphere equation of state.
129
D.Henderson
From equation (15) we can deduce that the result of the PY theory for the contact value of the
background function is
y(d) =
1 + η/2
(1 − η)2
. (17)
Remember that g(d+) = y(d) and c(d−) = −y(d) for PY/HS. Another result of interest is the
compressiblity, as given by the compressiblity equation,
(
1
kBT
∂p
∂ρ
)
(c)
=
(1 + 2η)2
(1 − η)4
. (18)
It is of interest to note that this is the square of a simple expression. We will return to this point
in the next section.
Also, it was found that at least through ρ2, the coefficients in the density expansion of the PY
DCF is a simple polynomial. Specifically,
c(R) = c0 + c1
R
d
+ c3
(
R
d
)3
. (19)
for R < d and zero for R > d. Presumably, it could be proven by induction that this expression is
valid to all orders in the density expansion. This has not been done. There is little incentive to do
this as the general validity of this expression has been established by general means that we shall
discuss in the following sections.
There are three unknowns in equation (19). Fortunately, we have three conditions to employ.
We know c(0) and, hence,
c0 = −
(1 + 2η)2
(1 − η)4
. (20)
Further, we know c(d−) and the integral of c(R). Hence,
c1 = 6η
(1 + η/2)2
(1 − η)4
(21)
and
c3 =
η
2
c0 . (22)
Thus, the DCF is completely determined.
It is worth noting that the pressure may be integrated to give the Helmholtz function, yielding
(
A − A0
NkBT
)
(p)
= 2 ln(1 − η) + 6
η
1− η
(23)
and
(
A − A0
NkBT
)
(c)
=
3η(1 − η/2)
(1 − η)2
− ln(1 − η), (24)
where A0 is the Helmholtz function of an ideal gas.
3. Fourier transform method
Baxter [9] has published an attractive method, based on the Fourier transform, for determining
the correlation functions. This appeared after the Laplace transform method that will be presented
in the next section. It is generally regarded as a superior method. The author does not share this
view. In his view, the Laplace transform method is more powerful and yields useful results that
are either not available from the Fourier transform method or have not been obtained by this
method. The basis of the author’s statement is that if the Laplace transform is known, the Fourier
130
Analytic methods for the Percus-Yevick hard sphere correlation functions
transform may be determined. However, the reverse is not true. Nonetheless, Baxter’s method is
pedagogically attractive.
The compressiblity is positive or zero and cannot be negative, in a thermodynamically stable
system. Thus, it is plausible that the compressiblity should be a simple square, as is the case with
equation (18). More generally, using the fact that the DCF vanishes for R > d, Baxter has shown
by a Wiener-Hopf factorization that
1 − ρ ˜c(k) = Q̃(k)2 = Q̃(k)Q̃(−k). (25)
The aforementioned observation that the compressiblity is a simple square is seen from equa-
tion (25) with k = 0. Additionally, this factorization permits the OZ equation, which involves a
three-dimensional convolution, to be written as two equations that each involve only one dimensi-
onal convolutions,
Rc(R) = −Q′(R) + 2πρ
∫ d
0
Q′(S)Q(S − R)dS (26)
and
Rh(R) = −Q′(R) + 2πρ
∫ d
0
(R − S)h(|R − S|)Q(S)dS, (27)
where Q(R) has the Fourier transform, Q̃(k), defined by
Q̃(k) = 1 − 2πρ
∫ d
0
exp(ikR)Q(R)dR. (28)
The function Q(R) is zero for R > d. The compressibility can be written in the form
β
∂p
∂ρ
= Q̃(0)2. (29)
Considering equation (27) for R < d yields
Q′(S) = aS + b (30)
with
a2 = c0 (31)
and
b = −
3η
2(1− η)2
. (32)
Thus,
Q(S) =
1
2
a(S2 − d2) + b(S − d). (33)
Equations (19)–(22) now follow from equation (26). This provides a justification for the as-
sumption in the preceeding section that the HS/PY DCF is a simple polynomial and the form
for the compressibility equation and pressure equation thermodynamics that was assumed in the
previous section.
It is worth noting that Baxter has shown that, in the PY theory, the pressure calculated from
(
pV
NkBT
)
(c)
= 1 − 4πρ
∫
∞
0
R2c(R)dR + 2πρ
∫
∞
0
R2e(R)[e(R) − 1]y2(R)dR
+
1
2π2ρ
∫
∞
0
k2 [ρc̃(k) + ln{1− ρc̃(k)}] dk (34)
gives the same pressure as that obtained from integrating ∂p/∂ρ from the compressibility equation
pressure. Equation (34) is valid for a general (simple) fluid. The derivation of equation (34) is not
straightforward but it is quite easy to verify equation (34) by differentiating this equation, using
131
D.Henderson
the PY approximation, and obtain equation (9). This equation has the advantage that the solution
of the PY equation need not be known for a sequence of states. This is no advantage for the HS
fluid, where a general analytic solution that is valid for all states is available. However, for systems
for which a numerical result must be obtained for each state, this is a real advantage especially
if there is a thermodynamically unstable region that must be crossed. Regretably, a further result
yielding the Helmholtz function is not available.
4. Laplace transform method
Thiele [7] and Wertheim [8] have obtained the PY/HS thermodynamics and correlation func-
tions using Laplace transform techniques. The method involves more algebra than the Fourier
transform method just discussed and has lead to what I feel is the incorrect conclusion that the
Fourier transform is to be preferred. My opinion is based on the fact that the Laplace transform
method yields both c(R) and g(R). The expression for c(R) is the analytic result that has been
obtained by the previous methods. In addition, an analytic expression for G(s), the Laplace trans-
form of g(R), is obtained. The inversion of G(s) is not analytic but useful explicit results for g(R)
can be obtained.
The Laplace transform F (s) of a function f(R) may, in the present context, be defined by
F (s) =
∫
∞
0
xf(x) exp(−sx)dx, (35)
where x = R/d. Thus,
C(s) =
∫
∞
0
xc(x) exp(−sx)dx −
∫
∞
1
xc(x) exp(−sx)dx. (36)
Wertheim shows quite rigorously that the PY/HS DCF is given by equations (19)–(22). However,
let us keep things simple here by assuming these results. Now, let us evaluate the first integral in
equation (36) by assuming that equation (19), with equations (20)–(22), is valid for all x.
Thus,
C(s) =
c0
s2
+
c1
s3
+
c3
s5
− exp(−s)
∫
∞
0
(t + 1)c(t + 1) exp(−st)dt. (37)
The second and third terms in equation (37) can be combined. Hence,
c1
s3
+
c3
s5
= −12η
L(s)L(−s)
(1− η)4s5
, (38)
where
L(s) = (1 + 2η) + (1 + η/2)s. (39)
Furthermore, the last term in equation (37) can be written in a form that involves L(s). After a
little algebra,
C(s) =
c0
s2
− 12η
L(s)L(−s)
(1 − η)4s5
− exp(−s)
L(s)S(−s)
(1 − η)4s5
, (40)
where
S(s) = −12η(1 + 2η) + 18η2s + 6η(1 − η)s2 + (1 − η)2s3. (41)
The polynomials L(s) and S(s) may be called Wertheim polynomials.
We are now in a position to obtain the Laplace transform of g(R). The OZ equation may be
written in the form
g(R12) − c(R12) =
(1 + 2η)2
(1 − η)4
+ ρ
∫
g(R13)c(R23)dr3 , (42)
132
Analytic methods for the Percus-Yevick hard sphere correlation functions
where equations (9) and (18) have been used. After a little algebra,
G(s) − C(s) = −
c0
s2
+
12η
s
G(s)[C(−s) − C(s)]. (43)
Solving for G(s) gives
G(s) =
−c0/s2 + C(s)
1 + 12η[C(s) + C(−s)]/s
. (44)
Using a little algebra and the identity,
(12η)2L(s)L(−s) − (1 − η)4s6 = S(−s)S(s), (45)
leads to the desired expression for G(s)
G(s) =
sL(s)
12ηL(s) + exp(s)S(s)
. (46)
This result for G(s) can be inverted explicitly [10,11] to yield results for g(R) for R 6 5d. Also,
this expression can be used to obtain a number of valuable results. For example, the contact value,
g(d+), can be obtained from the limit of sG(s) at large s. The result is equation (17). Also, we can
expand G(s) in powers of s.
sL(s)
12ηL(s) + exp(s)S(s)
=
1
s2
−
10− 2η + η2
20(1 + 2η)
+
(4 − η)(2 + η)
24(1 + 2η)2
s2 + · · · . (47)
Recognizing that
G(s) =
1
s2
+
∫
∞
0
xh(x) exp(−sx)dx, (48)
we obtain, by equating coefficients of like powers of s,
∫
∞
0
xh(x)dx = −
10− 2η + η2
20(1 + 2η)
(49)
and
∫
∞
0
x2h(x)dx = −
(4 − η)(2 + η2)
24(1 + 2η)2
. (50)
Equation (50) yields the PY/HS compressiblity that has been obtained previously. This process
can be used to obtain
∫
∞
0 xnh(x)dx for any n. Since the error in the PY g(R) is mostly in the
region of g(R) near d, these integrals become more accurate as n increases.
Alternatively, we can use equation (48) and obtain
∫
∞
0
xn+1h(x)dx = lim
s→0
[
(−1)n dnG(s)
dsn
−
(n + 1)!
sn+2
]
, (51)
for n > 1. Integrals involving x−n can be evaluated from
∫
∞
1
x−ng(x)dx =
1
n!
∫
∞
0
snG(s)ds, (52)
for n > 1. For these latter integrals, the value of the integral will become less accurate as n
increases. In the case where n is 6 and 12, these integrals can be used in a perturbation theory for
the Lennard-Jones potential.
This leaves the two integrals
∫
∞
0
h(x)dx =
∫
∞
0
[
G(s) −
1
s2
]
ds (53)
133
D.Henderson
and
∫
∞
1
h(x)
x
dx =
∫
∞
0
s
[
G(s) −
s + 1
s2
exp(−s)
]
ds, (54)
that are easily obtained from
∫
∞
0
xh(x) exp(−sx)dx = G(s) −
1
s2
(55)
and
∫
∞
1
h(x)
x
dx = G(s) −
s + 1
s2
exp(−s). (56)
Lebowitz [12] has applied the Laplace transform method to obtain corresponding PY expressions
for the direct correlation functions and Laplace transforms of the radial distribution functions and
the resulting thermodynamic functions for a mixture of hard spheres with additive diameters.
5. Summary
Results, drawn from the author’s lecture notes, for the solution of the PY equation for a hard
sphere fluid that have proven useful and that have appeared, in part, in the monograph of Gray and
Gubbins [5] are presented so as to make them more widely available. To the author’s knowledge,
only section 2 is original. The remainder of the article is a mini review, with simplifications, that
it is hoped will be of interest.
Acknowledgement
Jean-Pierre Hansen read a draft of this paper and made valuable suggestions for which the
author is grateful. However, any errors in this paper are the responsiblity of the author.
References
1. Barker J.A., Henderson D., J. Chem. Phys., 1967, 47, 2856, 4714; Rev. Mod. Phys., 1976, 48, 587.
2. Hansen J.-P., McDonald I.R. Theory of Simple Liquids, 3rd. Edition. Academic Press, London, UK,
2006.
3. Percus J.K., Yevick G.J., Phys. Rev., 1958, 110, 1.
4. Lebowitz J.L., Percus J.K., Phys. Rev., 1966, 144, 251.
5. Gray C.G., Gubbins K.E. Theory of Molecular Fluids. Volume 1: Fundamentals. Clarendon Press,
Oxford, UK, 1984, Appendix 5A.
6. Hutchinson P., Rushbrooke G.S., Physica, 1963, 29, 675.
7. Thiele E., J. Chem. Phys., 1963, 39, 474.
8. Wertheim M., Phys. Rev. Lett., 1963, 10, 321 (1963); J. Math. Phys., 1964, 5, 643.
9. Baxter R.J., Aust. J. Phys., 1968, 21, 563.
10. Smith W.R., Henderson D., Mol. Phys., 1970, 19, 411.
11. Smith W.R., Henderson D., Leonard P.J., Barker J.A., Grundke E.W., Mol. Phys., 2007, 106, 3.
12. Lebowitz J.L., Phys. Rev. A, 1964, 133, 895.
134
Analytic methods for the Percus-Yevick hard sphere correlation functions
Аналiтичнi методи для кореляцiйних функцiй твердих сфер у
наближеннi Перкуса-Євiка
Д.Гендерсон
Факультет хiмiї i бiологiї, Унiверситет Брайхем Янг, Прово, США
Отримано 18 травня 2009 р., в остаточному виглядi – 9 червня 2009 р.
Теорiя Перкуса-Євiка для твердих сфер дає можливiсть отримати дуже кориснi простi точнi вирази
для кореляцiйних функцiй. Тут представлено пiдсумковi нотатки лекцiй, що стосуються трьох ме-
тодiв отримання цих функцiй. Представленi нотатки є тiльки частково оригiнальними. Проте, вони
мiстять кориснi кроки i спрощення. Метою цiєї статтi є зробити цi нотатки бiльш доступними широ-
кому загалу.
Ключовi слова: Перкус-Євiк, твердi сфери, кореляцiйнi функцiї, iнтегральнi перетворення,
аналiтичнi результати
PACS: 02.30.Qy, 02.30.Rz, 05.20.Jj, 05.70.Ce, 64.30.+t
135
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