Cluster expansion for the description of condensed state: crystalline cell approach
A well-known cluster expansion, which leads to virial expansion for the free energy of low density systems, is modified in such a way that it becomes applicable to the description of condensed state of matter. To this end, the averaging of individual clusters over the states of an ideal gas is rep...
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irk-123456789-1576322019-06-21T01:29:21Z Cluster expansion for the description of condensed state: crystalline cell approach Bokun, G.S. Holovko, M.F. A well-known cluster expansion, which leads to virial expansion for the free energy of low density systems, is modified in such a way that it becomes applicable to the description of condensed state of matter. To this end, the averaging of individual clusters over the states of an ideal gas is replaced by the averaging over the states of a non-correlated crystal using single-particle cell potentials. As a result, we arrive at the expansion of the partition function in correlations on the basis of single-particle functions corresponding to the multiplicative approximation. The cell potentials defining these functions are found from the condition of the minimum of the remainder in the constructed decomposition. Широко вiдоме групове розвинення, яке приводить до вiрiального розвинення для вiльної енергiї розрiджених систем, модифiковано так, щоб його можна було застосовувати до конденсованого стану речовини. Для цього усереднення окремих кластерiв по станах iдеального газу замiнюється усередненням по станах некорельованого кристала, використовуючи комiрковi одночастинковi потенцiали. В результатi отримано розвинення статистичної суми по кореляцiях на базисi одночастинкових функцiй, якi вiдповiдають мультиплiкативному наближенню. Ґратковi потенцiали, що визначають вказанi функцiї, знаходяться з умови мiнiмiзацiї залишку в сконструйованому розвиненнi. 2018 Article Cluster expansion for the description of condensed state: crystalline cell approach / G.S. Bokun, M.F. Holovko // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43501: 1–16 . — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 5.20.-y, 61.72jd, 64.30.-t, 65.40.-b DOI:10.5488/CMP.21.43501 arXiv:1812.08536 http://dspace.nbuv.gov.ua/handle/123456789/157632 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
A well-known cluster expansion, which leads to virial expansion for the free energy of low density systems, is
modified in such a way that it becomes applicable to the description of condensed state of matter. To this end,
the averaging of individual clusters over the states of an ideal gas is replaced by the averaging over the states
of a non-correlated crystal using single-particle cell potentials. As a result, we arrive at the expansion of the
partition function in correlations on the basis of single-particle functions corresponding to the multiplicative
approximation. The cell potentials defining these functions are found from the condition of the minimum of the
remainder in the constructed decomposition. |
| format |
Article |
| author |
Bokun, G.S. Holovko, M.F. |
| spellingShingle |
Bokun, G.S. Holovko, M.F. Cluster expansion for the description of condensed state: crystalline cell approach Condensed Matter Physics |
| author_facet |
Bokun, G.S. Holovko, M.F. |
| author_sort |
Bokun, G.S. |
| title |
Cluster expansion for the description of condensed state: crystalline cell approach |
| title_short |
Cluster expansion for the description of condensed state: crystalline cell approach |
| title_full |
Cluster expansion for the description of condensed state: crystalline cell approach |
| title_fullStr |
Cluster expansion for the description of condensed state: crystalline cell approach |
| title_full_unstemmed |
Cluster expansion for the description of condensed state: crystalline cell approach |
| title_sort |
cluster expansion for the description of condensed state: crystalline cell approach |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157632 |
| citation_txt |
Cluster expansion for the description of condensed state: crystalline cell approach / G.S. Bokun, M.F. Holovko // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43501: 1–16
. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bokungs clusterexpansionforthedescriptionofcondensedstatecrystallinecellapproach AT holovkomf clusterexpansionforthedescriptionofcondensedstatecrystallinecellapproach |
| first_indexed |
2025-07-14T10:03:23Z |
| last_indexed |
2025-07-14T10:03:23Z |
| _version_ |
1837616231547404288 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 4, 43501: 1–16
DOI: 10.5488/CMP.21.43501
http://www.icmp.lviv.ua/journal
Cluster expansion for the description of condensed
state: crystalline cell approach
G.S. Bokun1, M.F. Holovko2
1 Belarusian State Technological University, 13a Sverdlov St., 220006 Minsk, Belarus
2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received May 2, 2018, in final form July 10, 2018
A well-known cluster expansion, which leads to virial expansion for the free energy of low density systems, is
modified in such a way that it becomes applicable to the description of condensed state of matter. To this end,
the averaging of individual clusters over the states of an ideal gas is replaced by the averaging over the states
of a non-correlated crystal using single-particle cell potentials. As a result, we arrive at the expansion of the
partition function in correlations on the basis of single-particle functions corresponding to the multiplicative
approximation. The cell potentials defining these functions are found from the condition of the minimum of the
remainder in the constructed decomposition.
Key words: lattice models, cluster expansions, single-particle cell potential, free energy
PACS: 5.20.-y, 61.72jd, 64.30.-t, 65.40.-b
1. Introduction
The cellular theory, which is based on the structuring of the states of the system, has made it possible
to solve a number of important problems in the physics of condensed state [1, 2]. The lattice version of this
theory has been used to address fundamental problems of statistical physics as well as to calculate specific
properties of various systems [3, 4]. The development and introduction of nanomaterials has driven these
approaches to be applied to the description of highly heterogeneous structures, which, in addition to large
gradients of order parameter fields, are characterized by a heterogeneous state accompanied by phase
transitions of various types [5, 6]. Here, as in the case of homogeneous systems, it seems effective to
involve mean-field representations based on the use of cell potentials [7, 8].
The present paper is devoted to the development of the basis of this approach. To this end, a system
of particles in the mean field of single-particle cell potentials is used as the reference system. The cell
potentials determine the average forces acting on a particle fixed in a selected cell from the particles
distributed in other molecular cells. Initially, only those states are taken into consideration where each
cell of the system is occupied by one molecule or is vacant. The heterogeneity in the system is taken into
account by the fact that the cell volumes are not the same. In order to calculate the partition function of
the initial condensed system, the perturbation theory is used based on the expansions over the states of
the reference system using generalized Mayer functions. These functions contain not only intermolecular
interaction potentials but also the necessary mean potentials. The latter are determined from the condition
for optimizing the deviation of the properties of the reference and the real systems. Furthermore, the
thermodynamic consistency of the theory is studied at the level of calculation of the first derivatives with
respect to thermodynamic parameters. The developed microscopic approach is generalized to the case of
long-range interaction. A possibility to take account of a more complete set of occupation number values
is discussed.
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
43501-1
https://doi.org/10.5488/CMP.21.43501
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
G.S. Bokun, M.F. Holovko
The paper is arranged as follows. In section 2, the lattice model and the description of a corresponding
reference system are presented. In order to take into account the inter-particle correlations in section 3,
we formulate the perturbation theory based on the expansions over the states of the reference system. In
section 4, we formulate the optimal choice of a single-particle cell potentials needed for the description
of the reference system. In section 5, the verification of thermodynamic self-consistency of the presented
approach is considered at the level of calculation of the first derivatives with respect to thermodynamic
parameters. In section 6, the generalization of the considered approach for the case of a more complete
set of occupation numbers values and the generalization to the case of the systems with long-range
interaction are discussed. We conclude in section 7.
2. The model and the reference system
In this paper we consider a lattice model with the Hamiltonian
HM =
1
2
M∑
i=1
Z∑
j(i)
Φ
(
qni , qn j
)
+
M∑
i=1
µini , (2.1)
where M is the total number of lattice sites (i.e., cells in the system),
∑Z
j(i) denotes consecutive summation
over all nodes surrounding the selected node i taking into account Z counted neighbors, µi is the chemical
potential value in the node i. The classification of states considered is similar to that adopted in [9]. The
variable qni determines the position of the particle (ni = 1i) or the vacancy (ni = 0i) in the cell i,
Φ
(
qni , qn j
)
is the interaction potential between two particles with coordinates qni and qn j . We suppose
that this potential is short-ranged enough and can be taken, for instance, in the Lennard-Jones form [10].
In order to describe the considered model with the Hamiltonian (2.1), we apply the concept of the
reference system widely used in the statistical theory of various condensed systems. Usually the reference
system is a simplified version of the real condensed system. It should include the main features of the real
model and should be described analytically with sufficient accuracy. For example, the simplest reference
system is the model of the ideal gas. The application of this model leads to the virial density expansion
for thermodynamic properties of real gases [11]. Another important reference system is the model of
hard spheres, which has been successfully employed in the modern liquid state theory [10]. In this paper,
to study the system with the Hamiltonian (2.1), we use the reference system that can be described by the
Hamiltonian
H0 =
M∑
i=1
µini +
M∑
i=1
Z∑
j(i)
φ j
(
qni
)
(2.2)
represented by single-particle cell potentials φ j
(
qni
)
, which has the meaning of the potential of an
external field whose source is located formally at the center of the cell j. φ j
(
qni
)
depends on the
variables that determine the position of particles or vacancies in the cell i and parametrically is a function
of the quantities characterizing the average distribution of particles or vacancies in the system and its
macroscopic state. In this case, n = 1 corresponds to the distribution of particles and n = 0 corresponds
to that of vacancies.
The connection between the Hamiltonian HM and H0 and the calculation of the single-particle
potentials is discussed in the next sections. In this section we consider only the description of the
reference system without elaboration of the single-particle cell potential φ j
(
qni
)
.
We consider the case of an inhomogeneous system with inhomogeneity characterized by the field of
mean occupation numbers
ρ0 = 〈ni〉0 . (2.3)
In order to simplify further expressions, we use abbreviationswherever possible, for exampleU(qni ) =
Uni denote the potential acting on a particle or a vacancy in the position qni . In order to take into account
the variable number of particles and vacancies in the system, we consider both q1i and q0i as two states
of a certain “virtual” particle. The first one corresponds to the position of a real particle and the second
one to that of a vacancy. In this case, it is possible to replace fixed particles by arbitrary ones.
43501-2
Cluster expansion for the description of condensed state: crystalline cell approach
This can be seen from the definition of ZM
ZM =
M∑
N=0
1
N!
QNe−µN . (2.4)
Due to indiscernibility of particles, N! is reduced, which makes it possible to represent Z (0)M of the
reference system in the form
Z (0)M =
1∑
n1=0
∫
ω1
dqn1 ...
1∑
ni=0
∫
ωi
dqni ...
1∑
nM=0
∫
ωM
dqnM exp
{
− β
[
M∑
l=1
(
µlnl +Unl
) ] }
, (2.5)
where β = 1/(kT), k is the Boltzmann constant, T is the temperature, ωi is the volume of the lattice
cell i,
Unl =
Z∑
j(l)
φ j
(
qnl
)
. (2.6)
In accordance with the representation (2.5), the particle distribution function over the volume of the
system turns out to be factorized and can be written in the form
D(0)M =
M∏
i=1
ρ
(
qni
)
, (2.7)
where
ρ
(
qni
)
= exp
[
−β
(
µini +Uni
) ]
. (2.8)
The normalization of this function, respectively, is represented by the expression
Z (0)M =
M∏
i=1
z0
i , (2.9)
where
z0
i =
1∑
ni=0
∫
ωi
exp
[
−β
(
µini +Uni
) ]
dqni , (2.10)
z0
i = Q0i + e
βµiQ1i , (2.11)
Qni =
∫
ωi
exp
[
− β
Z∑
j(i)
φ j
(
qni
) ]
dqni . (2.12)
Based on equations (2.7)–(2.12), we write the expression for the normalized distribution function,
which is necessary for averaging the magnitudes of the singlet, binary, and other types. Thus, denoting
the normalized functions by a cap above the notations of respective functions, we write
D̂M =
M∏
i=1
ρ̂
(
qni
)
, (2.13)
where
ρ̂
(
qni
)
=
exp
[
−β
(
µini +Uni
) ]
z0
i
. (2.14)
43501-3
G.S. Bokun, M.F. Holovko
We transform the above expressions based on the relation between µi and the mean values of the
occupation numbers ρ1i and ρ0i . Integrating (2.14) with respect to qni , we find
ρni =
eβµiniQni
z0
i
. (2.15)
From (2.15) it follows that
ρ1i
ρ0i
=
eβµiQ1i
Q0i
. (2.16)
The relation (2.16) allows us to write
eβµi =
ρ1iQ0i
ρ0iQ1i
. (2.17)
Substituting (2.17) into (2.11), we obtain
z0
i = Q0i +
ρ1iQ0i
ρ0i
=
Q0i
ρ0i
. (2.18)
The substitution of (2.17) and (2.18) allows us to write
ρ̂
(
qni
)
= ρni
exp
[
−β
Z∑
j(i)
φ j
(
qni
) ]
Qni
. (2.19)
Let us consider the averaging that employs (2.13), for example the characteristics of a binary type
L
(
qni , qn j
)
according to the definition
〈Li j〉0 =
1∑
n1=0
1∑
n2=0
...
1∑
nm=0
∫
ω1
ρ̂
(
qn1
)
...
∫
ω j
ρ̂
(
qn j
)
...
∫
ωM
ρ̂
(
qn j
)
...
∫
ωm
ρ̂ (qnm) L
(
qni qn j
)
. (2.20)
Transposing the summation in (2.20) and taking into account the independence of integration variables
and the normalized condition
1∑
ni=0
∫
ωi
ρ̂
(
qni
)
dqni =
1∑
ni=0
ρni = 1 (2.21)
we obtain
〈Li j〉0 =
1∑
ni=0
1∑
n j=0
∫
ωi
dqni
∫
ω j
dqn j L
(
qni , qn j
)
ρ̂
(
qni
)
ρ̂
(
qn j
)
, (2.22)
or
〈Li j〉0 =
1∑
ni=0
1∑
n j=0
Lni,n j ρni ρn j , (2.23)
Lni,n j =
∫
ωi
dqni
∫
ω j
dqn j L
(
qni , qn j
)
F̂11
(
qni
)
F̂11
(
qn j
)
. (2.24)
The functions F̂11 (q) in (2.24) correspond to singlet distribution functions F11 in the approximation
of the method of conditional distributions [9]
F̂11
(
qni
)
=
1
Qni
exp
[
− β
Z∑
j(i)
φ j
(
qni
) ]
. (2.25)
43501-4
Cluster expansion for the description of condensed state: crystalline cell approach
3. Perturbation theory
In this section, we consider a perturbation scheme for the treatment of the remainder
∆HM = HM − H0 =
1
2
M∑
i=1
Z∑
j(i)
∆φ
(
qni , qn j
)
, (3.1)
where
∆φ
(
qni , qn j
)
= Φ
(
qni , qn j
)
− φ j
(
qni
)
− φi
(
qn j
)
. (3.2)
Now, we represent the partition function of the original system in the form
ZM = Z (0)M 〈e
−β∆HM 〉0 , (3.3)
where 〈. . . 〉0 is the averaging represented by the expression
〈L〉0 =
1∑
n1=0
...
1∑
nM=0
∫
ω1
dqnU ρ̂
(
qnU
)
...
∫
ωM
ρ̂
(
qnM
)
dqnM L . (3.4)
To calculate (3.3), a cumulant expansion [10, 12] is used, leading to an expansion in powers of the
density if the averaging in (3.3) is performed over the states corresponding to an ideal gas [11]. In our
case, the distribution characteristic of an ideal crystal is used as the reference system. This allows us
to obtain a suitable description of the properties of a condensed system. The virial coefficients in this
case become density functions. In other words, the transformation (3.3) is an expansion over the cluster
correlations, although formally it has the form of an expansion over the Mayer functions. For the latter
we use the renormalized Mayer functions of the form
f
(
qni , qn j
)
= exp
[
− β∆φ
(
qni , qn j
) ]
− 1. (3.5)
We should note that for vacancies Φ(q0i , q0 j ) = Φ(q0i , q1 j ) = Φ(q1i , q0 j ) = 0. However, the Mayer
functions f (qni , qn j ) , 0 because in this case 4φ(qni , qn j ) , 0.
Using the procedure of group expansion in (3.3), we obtain
ZM = Z (0)M exp
[
1
2
M∑
i=1
Z∑
j(i)
1∑
ni=0
1∑
n j=0
∫
ωi
dqni
∫
ω j
dqn j ρ̂
(
qni
)
ρ̂
(
qn j
)
f
(
qni , qn j
)
+
1
6
M∑
i=1
Z∑
j(i)
Z∑
l(i)
1∑
ni=0
1∑
n j=0
1∑
nl=0
∫
ωi
dqni
∫
ω j
dqn j
∫
ωl
dqnl ρ̂
(
qni
)
ρ̂
(
qn j
)
ρ̂
(
qnl
)
× f
(
qni , qn j
)
f
(
qn j , qnl
)
f
(
qni , qnl
)
+ ...
]
. (3.6)
Or, due to representations (2.23) and (2.24),
ZM = Z (0)M exp
[
1
2
M∑
i=1
Z∑
j(i)
1∑
ni=0
1∑
n j=0
ρni ρn j fnin j
+
1
6
M∑
i=1
Z∑
j(i)
Z∑
l(i)
1∑
ni=0
1∑
n j=0
1∑
nl=0
ρni ρn j ρnl fnin jnl + ...
]
, (3.7)
43501-5
G.S. Bokun, M.F. Holovko
where
fnin j =
∫
ωi
dqni
∫
ω j
dqn j f
(
qni , qn j
)
F̂11
(
qni
)
F̂11
(
qn j
)
, (3.8)
fnin jnl =
∫
ωi
dqni
∫
ω j
dqn j
∫
ωl
dqnl f
(
qni , qn j
)
f
(
qn j , qnl
)
f
(
qni , qnl
)
× F̂11
(
qni
)
F̂11
(
qn j
)
F̂11
(
qnl
)
. (3.9)
From (3.7), we have the cluster expansion for the free energy
F = −kT ln ZM = −kT
[
ln Z0
M +
1
2
M∑
i=1
Z∑
j(i)
1∑
ni=0
1∑
n j=0
ρni ρn j fnin j
+
1
6
M∑
i=1
Z∑
j(i)
Z∑
l(i)
1∑
ni=0
1∑
n j=0
1∑
nl=0
ρni ρn j ρnl fnin jnl + ...
]
. (3.10)
4. An optimal choice of a single-particle potential
In order to apply the considered theory, we should specify the single particle potential. In this section
we propose an optimal choice for this potential. This approach is in some sense similar to the problem
of the connection between models with soft and hard core repulsions [13] which are successfully used
in modern liquid state theory [14]. The equation defining the single-particle potentials of the reference
system is determined from the self-consistent condition, which has several different formulations. One
of them is connected with the extremum of the remainder in the expansion (3.6) due to the fact that the
sum of all terms contained in (3.6) does not depend on the choice of single-particle potentials. Since a
change of the potentials leads to redistribution of contributions of individual terms, the best choice would
be the one with the maximum contribution of the terms that are taken into account. This is analogous to
the requirement of the minimum susceptibility of the system to a virtual external field and leads to the
condition
δ ln ZM
δφk(qnm )
= 0. (4.1)
Namely, the condition of the extremum of the part of the functional written in (3.7) corresponds
simultaneously to the condition that the sum in brackets in (3.7) tends to zero. Let us consider the proof
of the foregoing. We vary (3.7) over all the potentials in (2.6) assuming that they can all be independent of
each other, whichmakes it possible to substantially simplify the procedure of transformations. Performing
the variation with respect to an individual φk
(
qnm
)
and using (2.8) and (2.18), for the reference system
part we obtain
δ ln z0
m =
1
z0
m
{ ∫
ωm
δφk
(
q0m
)
exp
[
− β
Z∑
j(m)
φ j
(
q0m
) ]
dq0m
+
∫
ωm
δφ
(
q1m
)
exp
[
− β
Z∑
j(m)
φ j
(
q1m
) ]
eβµmdq1m
}
. (4.2)
Since the variation in δ ln ZM is satisfied for fixed µi , we transform the second term in ZM into a
form that contains explicit µi .
Im =
1
2
Z∑
j(m)
1∑
nm=0
1∑
n j=0
ρnm ρn j fnmn j =
1
2
Z∑
j(m)
1∑
nm=0
1∑
n j=0
∫
ωm
∫
ω j
exp
(
− β
{
Φ
(
qnm, qn j
)
43501-6
Cluster expansion for the description of condensed state: crystalline cell approach
+
∑
s,m, j
[
φs
(
qnm
)
+ φs
(
qn j
) ] })
eβµmnmeβµ jn j dqnmdqn j
1
z0
mz0
j
−
1
2
. (4.3)
Let us write the expression for the variation of (4.3) with respect to φk
(
qnm
)
δIm,k =
∑
j,k,m
1∑
nm=0
1∑
n j=0
∫
ωm
∫
ω j
δφk
(
qnm
)
exp
[
−βΦ
(
qnm, qn j
) ]
× exp
{
− β
∑
s,m, j
[
φs
(
qm
)
+ φs
(
qn j
) ] }
eβµmnmeβµ jn j dqnmdqn j
1
z0
mz0
j
−
Z∑
j(m)
1∑
nm=0
1∑
n j=0
∫
ωm
∫
ω j
exp
{
− βΦ
(
qnm, qn j
)
+
∑
s,m, j
[
φs
(
qnm
)
+ φs
(
qn j
) ] }
× eβµmnmeβµ jn j dqnmdqn j
1
z0
mz0
j
δ ln z0
m . (4.4)
The first sum on the right-hand side of equation (4.4) does not contain a term with j = m, which is
convenient to add and subtract, which allows the sum (4.2) and (4.4) to be represented in a form that
allows the separation of variables by cell numbers. Namely,
δ ln z0
m + δIm,k =
1
z0
m
1∑
nm=0
∫
ωm
δφk
(
qnm
)
exp
[
− β
Z∑
j(m)
φ j
(
qnm
) ]
eβµmnmdqnm
−
1
z0
mz0
k
1∑
nm=0
1∑
nk=0
∫
ωk
δφk
(
qnm
)
dqnm
∫
ωk
dqnk exp
[
−βΦ
(
qnm, qnk
) ]
× exp
{
− β
∑
s,m,k
[
φs
(
qnm
)
+ φs
(
qnk
) ] }
exp [β (µmnm + µknk)]
+
∑
j,m
Am, j = 0. (4.5)
In the relation (4.5), the term Am, j is the symmetrized part of δIm,k determined from equation (4.4).
So, for Am, j we can write
Am, j =
1
z0
mz0
j
1∑
nm=0
1∑
n j=0
eβ(µmnm+µ jn j )
∫
ωm
∫
ω j
[
δφk
(
qnm
)
− δ ln z0
m
]
× exp
[
−βΦ
(
qnm, qn j
) ]
exp
{
− β
∑
s,m, j
[
φs
(
qnm
)
+ φs
(
qn j
) ] }
dqnmdqn j . (4.6)
Subsequent separation of variables makes it possible to obtain an equation for the required potentials
φ j
(
qni
)
. Since both δφk
(
q1m
)
and δφk
(
q0m
)
are independent, after simplifications of equations (4.6),
we obtain a system of defining equations
exp
[
−βφk
(
qnm
) ]
=
1
z0
k
1∑
nk=0
eβµknk
∫
ωk
dqnk exp
{
− β
[
Φ
(
qnm, qnk
)
+
Z∑
s,m,k
φs
(
qnk
) ]}
. (4.7)
Equations (4.7) can be rewritten in another form, namely when the density field is used as a variable
that defines the system. Substituting (2.15) into (4.7) we arrive at a description
exp
[
−βφk
(
qnm
) ]
=
1∑
nk=0
ρnk
Qnk
∫
ωk
dqnk exp
{
− β
[
Φ
(
qnm, qnk
)
+
Z∑
s,m,k
φs
(
qnk
) ]}
. (4.8)
43501-7
G.S. Bokun, M.F. Holovko
Equations (4.7) and (4.8) define the single-particle cell potentials under the condition that the two-
vertex diagrams in the cluster expansion of the original partition function (3.6) are equal to zero. As
was shown in [15], this condition leads to the results that are equivalent to the quasichemical (or Bether-
Peierls) approximation. The inclusion of three-vertex diagrams in the equations (4.7) or (4.8) is also
possible but it would lead to more complicated equations and, therefore, we will not consider it here.
Now, we show that when equation (4.7) is satisfied, each term Aj in (4.5) turns out to be zero, which
indicates that the separation constant of the variables in (4.5) chosen to be zero is correct.
Thus, substituting (4.7) into (4.6), we obtain
Am, j =
1
z0
m
1∑
nm=0
eβµmnm
∫
ωm
[
δφk
(
qnm
)
− δ ln z0
m
]
exp
[
− β
∑
s,m
φs
(
qnm
) ]
dqnm . (4.9)
Using the definitions (2.11) and (2.12) in (4.9), we obtain
Am, j =
1
z0
m
1∑
nm=0
eβµmnm
∫
ωm
δφk
(
qnm
)
exp
[
− β
∑
s,m
φs
(
qnm
) ]
dqnm − δ ln z0
m . (4.10)
In turn, it is clear from (4.2) that the first sum in (4.10) is identical to the variation of the second
term, which proves that Aj = 0 when choosing single-particle potentials satisfying equation (4.7). As
already noted for specific calculations, (4.7) is preferable in the form (4.8) because this option allows
for an implicit relationship between the chemical potential and the density to be replaced by an explicit
one. Namely, for the chosen ρni the solution (4.8) is found with (2.12) taken into account. Then, due
to (2.17), the chemical potential and the free energy of the system are determined. To calculate the latter,
it is convenient to use the relation
−βF =
M∑
i=1
(
ln z0
i − ρ1i µi
)
. (4.11)
Taking into account equation (2.17), expression (4.11) is represented in the form
−βF =
M∑
i=1
[
ln z0
i − ρ1i ln
(
ρ1iQ0i
ρ0iQ1i
)]
. (4.12)
As a result, it follows from equation (4.7) that
1∑
n j=0
ρn j fnin j = 0. (4.13)
In order to prove (4.13), we multiply equation (4.8) by
ρnm
Qnm
exp
[
− β
∑
s,m
φs
(
qnm
) ]
. (4.14)
After integration, we obtain
1 =
1∑
nm=0
1∑
nk=0
ρnm ρnk
∫
ωm
∫
ωk
exp
[
−β∆φ
(
qnm, qnk
) ]
F̂11
(
qnm
)
F̂11
(
qnk
)
dqnmdqnk . (4.15)
The identity obtained with the definition of (3.8) proves the validity of equation (4.13). Hence, it
follows that when the potentials are determined from (4.8), the results of all three approaches are the
same. Due to (4.13), ln zi coincides with ln z0
i which is defined by the relation (2.18).
43501-8
Cluster expansion for the description of condensed state: crystalline cell approach
Substituting (2.18) into the formula (4.12), we obtain an expression for the free energy of the system
in the form
−βF =
M∑
i=1
(
ln Q0i − ln ρ0i + ρ1i ln ρ0i + ρ1i ln Q1i − ρ1i ln ρ1i − ρ1i ln Q0i
)
or
−βF =
M∑
i=1
(
ρ0i ln Q0i + ρ1i ln Q1i − ρ0i ln ρ0i − ρ1i ln ρ1i
)
. (4.16)
Such a representation of the right-hand side of equation (4.16) corresponds to a configurational
integral in the form
QN = Q0
N =
M∑
i=1
Q
ρ1i
1i Q
ρ0i
0i
ρ
ρ1i
1i ρ
ρ0i
0i
. (4.17)
5. Verification of thermodynamic self-consistency of the theory
The expression for Q0
N in the form (4.17) can be obtained using the Hamiltonian
H0
N =
M∑
i=1
Z∑
j(i)
φ j
(
qni
)
. (5.1)
To this end, it is necessary to consider the states corresponding to (5.1) based on the methods for
forming local equilibrium distributions [16]. We show that the conditions (4.8), when the contributions of
the third and the subsequent virial coefficients are not taken into account, give the identical ρi determined
by formulae
µi =
∂F
∂ρi
(5.2)
and
ρ1i =
∂ ln Z0
M
∂ (βµi)
. (5.3)
In order to verify the thermodynamic consistency of the theory, we first consider the validity of
equation (5.3). Using (2.18), we obtain
ρ1i =
∂ ln
(
Q0i + eβµiQ1i
)
∂ (βµi)
+
M∑
k,i
∂ ln z0
k
∂ (βµi)
=
1(
Q0i + eβµiQ1i
) [
eβµiQ1i +
∂Q0i
∂ (βµi)
+
∂Q1i
∂ (βµi)
]
+
M∑
k,1
∂ ln z0
k
∂ (βµi)
= ρ1i +
1(
Q0i + eβµiQ1i
) [
∂
∂ (βµi)
Q0i + e
βµi
∂Q1i
∂ (βµi)
]
+
M∑
k,i
∂ ln z0
k
∂ (βµi)
. (5.4)
It follows from (5.4) that (4.7) must satisfy the additional condition
∂
∂ (βµi)
Q0i + e
βµi
∂Q1i
∂ (βµi)
= 0. (5.5)
43501-9
G.S. Bokun, M.F. Holovko
Then, ρ1i calculated from the formulae (5.3) and (2.16) will coincide. To prove (5.5) we differentiate
the condition (4.7) with respect to βµm
∂
∂ (βµm)
[
−βφk
(
qnm
) ]
exp
[
−βφk
(
qnm
) ]
= −
1
z0
k
∂ ln z0
k
∂ (βµm)
1∑
nk=0
exp (βµknk)
×
∫
ωk
dqnk exp
{
− β
[
Φ
(
qnm, qnk
)
+
Z∑
s,m,k
φs
(
qnk
) ]}
+
1
z0
k
1∑
nk=0
exp (βµknk)
×
∫
ωk
[
Z∑
s=m,k
∂φs
(
qnk
)
∂ (βµm)
]
exp
{
− β
[
Φ
(
qnm, qnk
)
+
Z∑
s,m,k
φs
(
qnk
) ]}
dqnk . (5.6)
Multiplying (4.10) by
1
z0
m
exp
[
− β
∑
s,k,m
φs
(
qnm
) ]
, after integration over qnm we obtain
1
z0
m
∫
ωm
∂
∂ (βµm)
[
−βφk
(
qnm
) ]
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
dqnm
= −
1
z0
m
∂ ln z0
k
∂ (βµm)
∫
ωm
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
dqnm
+
1
z0
k
z0
m
1∑
nk=0
exp (βµknk)
∫
ωk
∫
ωm
dqnmdqnk exp
[
−βΦ
(
qnk , qnm
) ]
×
[
−β
Z∑
s,m,k
∂φs
(
qnk
)
∂ (βµm)
]
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
. (5.7)
Multiplying (5.7) by
1∑
nm
e−βµmnm and taking into account (2.11) and (4.7), we have
2
z0
m
1∑
nm=0
e−βµmnm
∫
ωm
∂
∂ (βµn)
[
−βφk
(
qnm
) ]
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
dqm
= −
∂ ln z0
k
∂ (βµm)
+
1
z0
k
1∑
nk=0
exp (βµknk)
∫
ωk
[
−β
Z∑
S,k
∂φs
(
qnk
)
∂ (βµm)
]
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
dqnk . (5.8)
As a result,
2
z0
m
1∑
nm=0
e−βµmnm
∫
ωm
∂
∂ (βµm)
[
−βφk
(
qnm
) ]
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
dqnm
=
∂ ln z0
k
∂ (βµm)
+
1
z0
k
1∑
nk=0
exp (βµknk)
∂Qnk
∂
(βµm) = 0. (5.9)
Performing the summation in (5.9) with respect to k , m, we find
1∑
nm=0
e−βµmnm
∂Qnm
∂ (βµm)
= 0. (5.10)
43501-10
Cluster expansion for the description of condensed state: crystalline cell approach
Likewise, we show that although in accordance with (4.7) each φs
(
qnk
)
depends on the chemical
potentials in the entire range of the k-th node, it turns out that
∂z0
m
∂ (βµl)
= 0 (5.11)
for l , m.
In combination with the condition (5.10), this will ultimately prove the thermodynamic consistency of
the theory when calculating the thermodynamic potentials F andΩ and their first derivatives, determined
by formulae (2.17) and (5.3). The proof of the condition (5.11) is carried out similarly to (5.5).
In order to reduce the transformations and make themmore transparent, we write a relation analogous
to (5.6) in the form{
∂
∂ (βµs)
[
−2βφk
(
qnm
) ] }
exp
[
−βφk
(
qnm
) ]
= −
∂ ln z0
k
∂ (βµs)
exp
[
−βφk
(
qnm
) ]
+
1
z0
k
1∑
nk=0
exp (βµknk)
∫
ωk
[
Z∑
s,m
∂βφs
(
qnk
)
∂ (βµs)
]
exp
{
− β
[
Φ
(
qnm, qnk
)
+
Z∑
s,m,k
φs
(
qnk
) ]}
dqnk . (5.12)
Let us apply the following operation to relation (5.12)
π̂ =
Z∑
k,m
1
z0
m
1∑
nm
exp (−βµmnm)
∫
ωm
exp
[
−β
Z∑
s,k
φs
(
qnm
) ]
dqnm ... (5.13)
writing equation (5.12) in a symbolic form
A = B + C. (5.14)
When converting the left-hand side of expression (5.14), the summation sign with respect to k , m
is put under the integration sign with respect to ωm, and taking into account (2.12), we write the result
in the form
π̂A = 2
1
z0
m
1∑
nm=0
exp (−βµmnm)
∂Qnm
∂ (βµs)
. (5.15)
Since in (5.15) s , m and µm and µs are independent variables, we switch the summation over nm
and differentiation with respect to µs, which due to (2.11) allows us to write
π̂A = 2
∂ ln z0
m
∂ (βµs)
. (5.16)
The result of the transformation of the expression B is equal to 1. It can be obtained if the operation (5.3)
is applied to exp
[
−βφk
(
qnm
) ]
taking the constant as the sign of operation (5.13). As a result, direct
application of π̂ gives
π̂B = −
∂ ln z0
k
∂ (βµs)
. (5.17)
To transform the expression C, we switch the summation and integration operations in (5.13), place
the operator π̂ under the integration sign with respect to ωk and take into account that according to (4.7)
π̂ exp
{
− β
[
Φ
(
qnk , qnm
) ] }
= exp
[
−βφm
(
qnk
) ]
. (5.18)
43501-11
G.S. Bokun, M.F. Holovko
Then, π̂C becomes of the form
π̂C =
1
z0
k
1∑
nk=0
exp (βµknk)
∂ ln Qk
∂ (βµs)
. (5.19)
Taking (2.11) into account, condition (5.19) gives
π̂C =
∂ ln z0
k
∂ (βµs)
. (5.20)
The results (5.17) and (5.20) show that according to (5.14), the expression (5.16) is identically equal
to zero. This, in turn, proves the validity of (5.12).
Similarly to the consistency of (5.3) and (2.15) proved by derivations (5.4)–(5.20), one can show
that (5.2) is consistent with (2.16). To this end, it is necessary to perform transformations similar to
those performed for (4.7) over (4.8), differentiating each of the equations (5.4)–(5.20) with respect to
thermodynamic variables ρni but not as it was beforewith respect to µs. Further analysis of the consistency
problem involves comparing the results of calculating the second derivatives of thermodynamic potentials,
and will be considered separately.
6. Discussion
In this paper, for a lattice model with the Hamiltonian (2.1) and a pair interaction potentialΦ(qni , qn j )
we formulate a reference system with the Hamiltonian (2.2) and a single-particle cell potential φ j(qni ).
The potential φ j(qni ) can be interpreted as themean potential exerted by a particle in the lattice cellωj on a
particle in the lattice cellωi . It is shown that the systemwith the single-cell potential φ j(qni ) reduces to the
description of a Fermi-like lattice model in an external field. Using this system as the reference system and
renormalizedMayer functions in the form (3.5) with∆φ(qni , qn j ) in the form (3.2), the generalized cluster
expansion for the free energy of the considered system is obtained. The cell potentials are calculated
from the condition of the minimum difference of thermodynamic properties of the systems with the
Hamiltonians (2.1) and (2.2). Such a procedure is considered under the condition that the two-vertex
diagrams in the cluster expansion of the partition function for the system with the Hamiltonian (2.1)
are equal to zero. As a result, for the single-particle cell potential, a system of equations (4.7) was
obtained. This system can also be presented in the form of equation (4.8). As it was noted previously
in the reference [15], such a description is equivalent to the quasichemical approximation. We should
note that the descriptions of the considered system in the framework of the Hamiltonians (2.1) and (2.2)
are exactly equivalent only in the case when all the terms in the cluster expansion (3.10) are taken into
account at the calculation of the single-particle potentials. There are two principal differences between
the considered equation (4.7) and the corresponding equations in traditional approaches such as the
mean field approximation formulated in the framework of the field theoretical approach [17, 18] or the
density functional approach [19] well developed for non-lattice fluid systems. The first difference is
connected with the presence of the interparticle potential Φ(qni , qn j ) in equation (4.7) in the exponential
form. The second one is connected with the inequality s , m, k in the exponent of equation (4.7). It
means that instead of the singlet distribution function F̂11(qnk ), which usually appears in the mean field
approximation, in the approach considered, the function F̂11(qnk ) exp
[
βφm(qnk )
]
appears. This is the
result of the peculiarity of the Mayer functions in the form (3.5) with ∆φ(qni , qn j ) in the form (3.2). If
we neglect the condition s , m, k due to the renormalization condition (2.21), the equation (4.7) can be
rewritten in the form
e−βφk (qnm ) = 1 +
1∑
nk=0
ρnk
∫
dqnk
{
exp
[
−βΦ
(
qnm, qnk
) ]
− 1
}
F̂11(qnk ) , (6.1)
43501-12
Cluster expansion for the description of condensed state: crystalline cell approach
which after linearization of the exponents exp
[
−βφk(qnm )
]
and exp
[
−βΦ
(
qnm, qnk
) ]
leads to a tradi-
tional form for the mean field approximation
φk(qnm ) =
1∑
nk=0
ρnk
∫
dqnkΦ
(
qnm, qnk
)
F̂11
(
qnk
)
. (6.2)
The equation (4.7) describes the single-particle cell potentials φk(q1m ) for real particles, but for
vacancies it is more of a problem. This problem is similar to the description of solvophobic interaction
in the theory of solutions [20, 21] and for a correct description of single-particle cell potential φk(q0m )
for vacancies, at least three-vertex diagrams in the equation (4.7) should be included. In the two-
vertex diagram approximation for φk(q0m ) due to inequality s , m, k there appears only some constant
corresponding to the change of the chemical potential due to the creation of a vacancy.
The theory presented here is easily generalized when it is necessary to take into account a larger
number of possible states. This is achieved by expanding the possible values of the occupation numbers,
when ni = 0, 1, 2 . . . . In this case, for everymicroconfiguration given by the set of values {n1, . . . , nm}, the
expression for the Hamiltonian is completely preserved in the form (2.1). So, for example, for n = 0, 1, 2
in all formulae (2.2)–(5.20) one should remember that
Φ
(
qni , qn j
)
=
0, ni, nj = 0 ,
h
(
qi, qj
)
, ni, nj = 1 ,
h
(
qi, qj
)
+ h
(
qi, q′j
)
, ni, nj = 2 ,
h
(
qi, qj
)
+ h
(
qi, q′j
)
+ h
(
q′i, qj
)
+ h
(
q′i, q
′
j
)
, ni, nj = 4 .
(6.3)
Here, h(qi, qj) is the intermolecular interaction potential of two particles in positions qi and qj , when
(qi, q′i ) ∈ ωi , (qj, q′j) ∈ ωj .
In addition to taking into account (6.3) in all expressions (2.2)–(5.20), the summation over n = 0, 1
must be extended to the case n = 0, 1, 2, additionally taking into account that∫
ωi
dq0i =
1
ωi
∫
ωi
dqi,
∫
ωi
dq1i =
∫
ωi
dqi,
∫
ωi
dq2i =
∫
ωi
dqi
∫
ωi
dq′i . (6.4)
Thus, it is shown that the properties of a condensed system can be described by combining the model
of an ideal crystal with a group expansion over the modified Mayer functions that impose correlations on
the properties of an ideal crystal.
It seems justified to extend the developed approach to take into account long-range effects in essentially
inhomogeneousmedia. Let us demonstrate the possibility of such propagation using the example when the
inhomogeneity of the medium is described by different sizes of microcells, provided that in each of them
there is one particle. Representing the energy of the system by short-range Φ(i, j) and long-range V(i, j)
potentials of pair interactions, respectively, for particles in positions qi , qj , we present the configurational
integral QN of the system in the form
QN = Q0
N
〈
exp
[
− β
N∑
i< j
V(i, j)
] N∏
i< j
[
1 + f (i, j)
]〉
0
, (6.5)
where
f (i, j) = exp
{
− β [Φ (i, j)] − φ j (i) − φi ( j)
}
− 1 , (6.6)
Q0
N =
N∏
i=1
Qi , Qi =
∫
vi
exp
[
− β
∑
k,i
φk (i)
]
dqi . (6.7)
Q0
N is the configurational integral of an ideal crystal, expressed through single-particle cell potentials of
mean forces φ j (i), f (i, j) is a renormalizedMayer function, the angle brackets 〈...〉0 denote averaging over
43501-13
G.S. Bokun, M.F. Holovko
the equilibrium states of the reference system. The subsequent cumulant expansion of the expression (2.2)
with respect to the functions (2.3) allows one to write in the approximation of the second virial coefficient
ln QN = ln Q0
L + ln Q0
N +
∑
i, j
〈 f (i, j)g(i, j)〉0 + ... , (6.8)
Q0
L =
〈
exp
[
− β
∑
i< j
V(i, j)
]〉
0
. (6.9)
The averaging in relations (6.7) and (6.9) is realized by multiplying the unary distribution functions
F11(i) and F11( j)
F11 (i) =
1
Qi
exp
[
− β
∑
k,1
φk(i)
]
. (6.10)
A special feature of relation (6.9) is that the renormalized Mayer function f (i, j) is modulated here
by a binary function, for which a consistent calculation scheme was developed in [10]. Applying the
procedure of self-consistent calculation of the potentials φ j (i) in accordance with the foregoing, we
arrive at a closed system of equations of the form
exp
[
−βφ j (i)
]
=
1
Q j
∫
vj
g(i, j) exp [−βΦ (i, j)] × exp
[
− β
∑
k,i, j
φk ( j)
]
d j . (6.11)
The system of equations (6.11) differs from the one used earlier because its kernel, in addition to
the point short-range potential, contains a binary function for a system of particles with a Coulomb
interaction, the expression for which has the form [10]
F2(i, j) = F11(i)F11( j)g(i, j) ,
g(i, j) = exp[−βu(i, j)] , (6.12)
where u(i, j) is a screening potential. Such a form solves the problem of the divergence of the integrals
when calculating the free energy (6.7). As a result, long-range effects apply also to the renormali-
zation of single-particle cell potentials. In expression (6.12), only the initial terms of the series are
written corresponding to the Debye description of ion systems. A more complete representation for rela-
tion (6.12) follows when calculating additional terms of the distribution function g(i, j) using collective
variables [10]. A generalization of Debye screening to the system of mobile ions in lattice models was
recently discussed in [4]. It leads to the change of the traditional inverse Debye length κ to the new one
κ =
[
βe2c (1 − c) /
(
ε0εh3) ]1/2, where e is the charge of a mobile ion, ε0 is the dielectric permittivity of
the vacuum, ε is the relative dielectric permittivity of the medium, c is the concentration of mobile ions,
h is the length of the cell.
The peculiarity of the equations in the case under consideration is due to the fact that the function
g(i, j) is determined by averaging not over the states of an ideal gas, but over the states of an ideal crystal.
Accordingly, for example for i = 1, j = 2, the function g(i, j) is defined by an expression of the form
g(1, 2) =
∫
v3
...
∫
vN
exp
[
− β
N∑
l<m=1
V (l,m)
]
F11 (3) F11 (4) ...F11 (N) d3d4...dN . (6.13)
There are reasons to believe that the appearance of additional Gaussian in (6.13) introduced by
the functions F11 will lead to an improvement in the convergence of the series (6.12) obtained in the
framework of the collective variablesmethod [10] since a similar procedurewas successful in constructing
the description using an effective potential [22].
43501-14
Cluster expansion for the description of condensed state: crystalline cell approach
7. Conclusions
A method for modifying the cluster virial expansion, which makes it possible to describe condensed
state, is proposed. The method is based on replacing the averaging over the states of an ideal gas by
the averaging over the states of an ideal crystal. The approach outlined differs from the well-known
perturbation theory because the basis with respect to which the expansion is performed is not a part
of the original Hamiltonian but is introduced independently. The basis distribution conveys the main
features of the solid state of matter, where the motion of molecules is of an oscillatory nature with
respect to the lattice sites. As a result, the Hamiltonian of the basic reference system is represented
by the sum of single-particle cell mean force potentials. It is shown how the Mayer functions can be
modified so that they would act as a small parameter for the subsequent expansion of the thermodynamic
potential by cumulant expansions. From the condition of independence of the initial partition function
on the introduced potentials of the mean forces, a system of integral equations determining the above
potentials is obtained. The expression for the free energy functional is obtained with the two first terms
of its expansion in correlations taken into account. It is shown that the obtained equations satisfy several
optimization conditions for the parameters characterizing the properties of the reference system and the
initial system. The thermodynamic consistency of various methods for calculating the thermodynamic
characteristics of a condensed medium both in the canonical and in the grand canonical ensembles is
proved. A possibility of using the developed approach to take into account not only short-range but also
long-range interactions is shown.
Acknowledgements
This project has received funding from European Unions Horizon 2020 research and innovation
programme under theMarie Skłodowska-Curie (grant agreement No 734276), the Belarusian Republican
Foundation for Fundamental Research (grant NoΦ16K-061) and the State Fund for Fundamental Research
of Ukraine (grant No Φ73/26-2017). We thank Ivan Kravtsiv and Dung di Caprio for the careful reading
of the manuscript and useful comments.
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Групове розвинення для опису конденсованих систем:
пiдхiд кристалiчних комiрок
Г.С. Бокун1,М.Ф. Головко2
1 Бiлоруський державний технологiчний унiверситет, вул. Свердлова, 13а, 220006Мiнськ, Бiлорусь
2 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна
Широко вiдоме групове розвинення, яке приводить до вiрiального розвинення для вiльної енергiї роз-
рiджених систем, модифiковано так,щоб його можна було застосовувати до конденсованого стану речо-
вини. Для цього усереднення окремих кластерiв по станах iдеального газу замiнюється усередненням по
станах некорельованого кристала, використовуючи комiрковi одночастинковi потенцiали. В результатi
отримано розвинення статистичної суми по кореляцiях на базисi одночастинкових функцiй, якi вiдповiда-
ють мультиплiкативному наближенню. Ґратковi потенцiали,що визначають вказанi функцiї, знаходяться
з умови мiнiмiзацiї залишку в сконструйованому розвиненнi.
Ключовi слова: ґратковi моделi, груповi розвинення, одночастинковий ґратковий потенцiал, вiльна
енергiя
43501-16
https://doi.org/10.1080/00268976.2010.547524
https://doi.org/10.1063/1.4921242
https://doi.org/10.1016/0009-2614(77)85161-0
https://doi.org/10.1016/j.molliq.2018.01.015
Introduction
The model and the reference system
Perturbation theory
An optimal choice of a single-particle potential
Verification of thermodynamic self-consistency of the theory
Discussion
Conclusions
|