Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction
The kinetic theory of mixtures with a multistep potential (MSP) is considered. A passage to the limit of the smooth continuous potential for the kinetic equation and the potential energy density balance equation is analyzed. When the “hard spheres + soft tail” form for the limiting potential is chos...
Saved in:
| Date: | 2010 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Відділення фізики і астрономії НАН України
2010
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/13437 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction / Y.A. Humenyuk, M.V. Tokarchuk // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 450-457. — Бібліогр.: 26 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-13437 |
|---|---|
| record_format |
dspace |
| spelling |
Humenyuk, Y.A. Tokarchuk, M.V. 2010-11-08T17:31:55Z 2010-11-08T17:31:55Z 2010 Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction / Y.A. Humenyuk, M.V. Tokarchuk // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 450-457. — Бібліогр.: 26 назв. — англ. 2071-0194 PACS 05.20.Dd, 05.60.Cd, 51.10.+y https://nasplib.isofts.kiev.ua/handle/123456789/13437 The kinetic theory of mixtures with a multistep potential (MSP) is considered. A passage to the limit of the smooth continuous potential for the kinetic equation and the potential energy density balance equation is analyzed. When the “hard spheres + soft tail” form for the limiting potential is chosen, the kinetic equation reduces to that of the kinetic variational theory (KVT), while the limiting balance equation for the kinetic energy density differs from its KVT counterpart. Розглянуто кiнетичну теорiю сумiшей з багатосходинковим потенцiалом взаємодiї. Проаналiзовано граничний перехiд до плавного неперервного потенцiалу для кiнетичного рiвняння i рiвняння балансу для густини потенцiальної енергiї. Коли граничний потенцiал вибрано у формi “твердi кульки + плавний хвiст”, кiнетичне рiвняння зводиться до рiвняння кiнетичної варiацiйної теорiї (КВТ), однак граничне рiвняння балансу для густини кiнетичної енергiї вiдрiзняється вiд вiдповiдного рiвняння КВТ. en Відділення фізики і астрономії НАН України Загальні питання теоретичної фізики Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction Гранична поведінка кінетичної теорії для систем із багатосходинковим потенціалом Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction |
| spellingShingle |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction Humenyuk, Y.A. Tokarchuk, M.V. Загальні питання теоретичної фізики |
| title_short |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction |
| title_full |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction |
| title_fullStr |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction |
| title_full_unstemmed |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction |
| title_sort |
limiting behavior of the kinetic theory for systems with multistep interaction |
| author |
Humenyuk, Y.A. Tokarchuk, M.V. |
| author_facet |
Humenyuk, Y.A. Tokarchuk, M.V. |
| topic |
Загальні питання теоретичної фізики |
| topic_facet |
Загальні питання теоретичної фізики |
| publishDate |
2010 |
| language |
English |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Гранична поведінка кінетичної теорії для систем із багатосходинковим потенціалом |
| description |
The kinetic theory of mixtures with a multistep potential (MSP) is considered. A passage to the limit of the smooth continuous potential for the kinetic equation and the potential energy density balance equation is analyzed. When the “hard spheres + soft tail” form for the limiting potential is chosen, the kinetic equation reduces to that of the kinetic variational theory (KVT), while the limiting balance equation for the kinetic energy density differs from its KVT counterpart.
Розглянуто кiнетичну теорiю сумiшей з багатосходинковим потенцiалом взаємодiї. Проаналiзовано граничний перехiд до плавного неперервного потенцiалу для кiнетичного рiвняння i рiвняння балансу для густини потенцiальної енергiї. Коли граничний потенцiал вибрано у формi “твердi кульки + плавний хвiст”, кiнетичне рiвняння зводиться до рiвняння кiнетичної варiацiйної теорiї (КВТ), однак граничне рiвняння балансу для густини кiнетичної енергiї вiдрiзняється вiд вiдповiдного рiвняння КВТ.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13437 |
| citation_txt |
Limiting Behavior of the Kinetic Theory for Systems with Multistep Interaction / Y.A. Humenyuk, M.V. Tokarchuk // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 450-457. — Бібліогр.: 26 назв. — англ. |
| work_keys_str_mv |
AT humenyukya limitingbehaviorofthekinetictheoryforsystemswithmultistepinteraction AT tokarchukmv limitingbehaviorofthekinetictheoryforsystemswithmultistepinteraction AT humenyukya graničnapovedínkakínetičnoíteoríídlâsistemízbagatoshodinkovimpotencíalom AT tokarchukmv graničnapovedínkakínetičnoíteoríídlâsistemízbagatoshodinkovimpotencíalom |
| first_indexed |
2025-11-24T04:30:54Z |
| last_indexed |
2025-11-24T04:30:54Z |
| _version_ |
1850842154615701504 |
| fulltext |
GENERAL PROBLEMS OF THEORETICAL PHYSICS
450 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
LIMITING BEHAVIOR OF THE KINETIC THEORY
FOR SYSTEMS WITH MULTISTEP INTERACTION
Y.A. HUMENYUK,1 M.V. TOKARCHUK1, 2
1Institute for Condensed Matter Physics, Nat. Acad. of Sci. of Ukraine
(1, Svientsitskii Str., Lviv 79011, Ukraine; e-mail: josyp@ ph. icmp. lviv. ua )
2National University “Lvivska Politekhnika”
(12, Bandera Str., Lviv 79013, Ukraine; e-mail: mtok@ ph. icmp. lviv. ua )
PACS 05.20.Dd, 05.60.Cd,
51.10.+y
c©2010
The kinetic theory of mixtures with a multistep potential (MSP)
is considered. A passage to the limit of the smooth continuous
potential for the kinetic equation and the potential energy den-
sity balance equation is analyzed. When the “hard spheres + soft
tail” form for the limiting potential is chosen, the kinetic equa-
tion reduces to that of the kinetic variational theory (KVT), while
the limiting balance equation for the kinetic energy density differs
from its KVT counterpart.
1. Introduction
Triple and higher-order collisions present a consider-
able difficulty for the development of the kinetic the-
ory for dense systems with realistic potentials, e.g., of
the Lennard-Jones type and, specifically, for treating a
long-range interaction in explicit way. The significant
progress has been achieved, therefore, for model systems
based on the Enskog kinetic theory for hard spheres [1–4]
which include the long-range interaction in some approx-
imate way. The addition of a Fokker–Planck-type term
to the Enskog collision integral [5] results in that the
hard spheres move as Brownian particles between hard-
core collisions. However, the shear viscosity and thermal
conductivity coefficients of the model differ appreciably
[6] from the molecular dynamic data.
Using the approach of maximizing the entropy with
certain constraints, several kinetic mean-field theories
(KMFT) for a system of particles interacting via a “hard-
core + soft tail” potential were proposed [7–9]. They
take the attraction into account through the mean-field
collision integral and thermodynamic quantities. A mix-
ture version was considered, and the theory in the Kac-
tail limit was examined [10]. Though the proposed bal-
ance equation for the kinetic energy density is of non-
typical form, the theory is considered successful up till
now [11]. There also appear the semiphenomenological
ways to treat the interparticle attraction through, e.g.,
the average cross-section of the momentum transfer used
for viscosity [12].
The kinetic theory for MSP [13–17] generalizes the
theory for a square well [18–20]. The both are char-
acteristic in that the long-range attraction is included
into an appropriate collision integral explicitly and in
irreversible fashion. The balance equation for the inter-
action energy density is a necessary ingredient of these
theories. It is worth noting that one can deduce the sim-
plest version of KMFT, called KVT, by passing to the
limit upon the potential, when the starting point is the
MSP kinetic theory, the number of steps increases, and,
simultaneously, separations between them decrease [14].
Thus, we get a way to verify KVT by the MSP theory.
The question on the limiting behavior concerning the
MSP kinetic equation alone was considered earlier [14],
but for a one-component model and for such a closure re-
lation [18] for the two-particle distribution function that
the potential energy equation was neglected. It was re-
vealed that the MSP kinetic equation transforms in the
limit to the KVT-type one, while the question on the lim-
iting behavior of the balance equations for kinetic and
potential energy densities remained without attention.
We consider a mixture version of the MSP kinetic the-
ory, when the closure relation used takes the potential
energy density into consideration [19]. The passage to
the limit is analyzed both on the kinetic level and in
hydrodynamic equations. The limiting equation for the
LIMITING BEHAVIOR OF THE KINETIC THEORY
kinetic energy density is found to differ from that pro-
posed in KVT and to include contributions to the stress
tensor and the heat flux from the soft tail of a limiting
potential. In Section 2, we present the equations of the
kinetic level, the closure, and the corresponding hydro-
dynamics [21]. In Section 3, the main ideas and results
of KVT are outlined. The passage to the limit of a con-
tinuous potential for both the kinetic and hydrodynamic
equations is considered in Section 4. In the last section,
conclusions are made.
2. Kinetic Description and Hydrodynamic
Equations
The contribution from a long-range interaction to the to-
tal energy density cannot be neglected at intermediate
and high number densities. This is the reason for the
kinetic equation for the one-particle distribution func-
tion to be complemented with an appropriate balance
equation for the potential energy density [19, 20, 22, 23].
This circumstance changes qualitatively the scenario of
the kinetic stage of evolution as compared with that of
the hard-sphere system and must be included into the
construction of kinetic theories for dense systems with a
realistic smooth interaction [24, 25].
We consider an M -component system of classical par-
ticles interacting by pair central forces. The multistep
potential φMS
ij mimics the realistic one consisting of a
hard core and a set of repulsive and attractive walls of
finite heights. The geometry of MSP is determined by
the following parameters [15,16,21]: σ0
ij , σrl
ij , σal
ij are allo-
cations of the hard-core, repulsive, and attractive walls,
Kr
ij and Ka
ij are numbers of the repulsive and attrac-
tive walls (Fig. 1). Parameters φrl
ij and φal
ij denote values
of MSP between walls, while εrlij = φrl
ij − φ
r,l+1
ij , εalij =
φal
ij − φ
a,l−1
ij characterize heights of the walls and are in-
troduced in such a way that εrlij , εalij > 0. The plateaux
are numbered in the direction of increasing rij : from
the hard core for the repulsive part and from the first
attractive wall for the attractive part of MSP.
The one-particle distribution functions f i1 are gov-
erned by the system of kinetic equations [21]
[∂t + vi · ∇] f i1(r,vi, t) =
M∑
j=1
{
IE
ij [f
ij
2 ] + IMS
ij [f ij2 ]
}
, (1)
where ∂t ≡ ∂
∂t , ∇ ≡
∂
∂r , and f ij2 are the two-particle dis-
tribution functions, the closure relations for which will
be given below. The collision integrals IE
ij and IMS
ij de-
scribe pair processes at the hard core and the walls of
Model multistep potential φMS
ij , with Kr
ij = 2, Ka
ij = 5
finite height. Expression for the Enskog-type term, IE
ij ,
is conventional [4, 21] and not given here.
The contribution IMS
ij can be presented in a compact
form [21] with the aid of a type-of-step parameter q and
a type of process p at the step: q = ‘r’ for the repulsive
steps and q = ‘a’ for the attractive ones. There are three
types of (ij)-processes at a step: descending p = ⊕,
ascending p = , and reflection p = ⊗, therefore
IMS
ij [f ij2 ] =
r,a∑
q
Kq
ij∑
l=1
⊕, ,⊗∑
p
Iqlpij [f ij2 ]. (2)
We ascribe formal numerical values to the parameters as
follows:
q =
(
r
a
)
=
(
−1
+1
)
, p =
⊕
⊗
=
+1
−1
0
, (3)
so that contributions to the IMS
ij can be given as
Iql⊗ij [f ij2 ] = (σqlij )
2
∫
dvjdσ̂ vjiσ θ(vjiσ) θ(v
ql
ij − vjiσ)×
×[f ij2 (r,v′i, r− qσ
ql
ij ,v
′
j)
−q −
−f ij2 (r,vi, r + qσqlij ,vj)
−q] (4)
for the reflection process at step {q, l} and
Iqlpij [f ij2 ] = (σqlij )
2
∫
dvjdσ̂ vjiσ θ(vjiσ +
p− 1
2
vqlij )×
×[f ij2 (r,vqlpi , r + qpσqlij ,v
qlp
j )−qp −
−f ij2 (r,vi, r− qpσqlij ,vj)
+qp] (5)
for the descending and ascending processes. Here, nu-
merical values of p and q are used to determine the po-
sition of particle j for the function f ij2 , the argument
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 451
Y.A. HUMENYUK, M.V. TOKARCHUK
of the θ function in Eq. (5), and to fix the right or left
limiting values for f ij2 in Eqs. (4), (5). Otherwise, q and
p are symbols being used for designation.
Notations introduced for velocities of particles i and j
mean: vi, vj – just before a process; v′i, v′j – just after
a pair collision at the hard core or a reflection process
at a step; vqlpi , vqlpj – after processes p = ⊕, at the
step {q, l}; σ̂ is a unit vector from the center of particle
j to the center of particle i, σ0
ij = σ0
ij σ̂, σqlij = σqlij σ̂,
vjiσ = (vj −vi)· σ̂. The corresponding rules for the pair
processes read
v′i = vi + 2Mjivji · σ̂σ̂, v′j = vj − 2Mijvji · σ̂σ̂, (6)
vqlpi = vi +Mji[vjiσ −
√
v2
jiσ + p(vqlij )2] σ̂,
vqlpj = vj −Mij [vjiσ −
√
v2
jiσ + p(vqlij )2] σ̂, (7)
where Mji = mj/(mi + mj), v
ql
ij = (2εqlij/µij)
1/2 is the
step height in velocity units, µij = mimj/(mi + mj) is
the reduced mass. The quantity f ij2 is discontinuous in
configurations, for which MSP is, and f ij2 (.)± denotes
the right (+) or left (−) limiting value.
Averages with f i1 and f ij2 will be designated as
〈ψ1〉1,ivi
df=
∫
dvi f i1 ψ1, 〈ψ2〉2,ijvixj
df=
∫
dvidxj f
ij
2 ψ2,
where the subscripts indicate variables of integration.
The balance equation for the potential energy density
ep(r, t) df=
M∑
i,j=1
〈1
2
φMS
ij (rij)〉2,ijvixj
∣∣
ri→r
(8)
must be considered on the kinetic level of description
together with the kinetic equations for f i1. Its heuristic
derivation based on ideas of the numbers of direct and
inverse collisions gives [21]
∂te
p +∇· [Vep + qp] = sp, (9)
where V(r, t) is the hydrodynamic velocity defined below
Eq. (17) below and
qp(r, t) =
M∑
i,j=1
〈1
2
ciφMS
ij (rij)〉2,ijvixj
∣∣
ri→r
, (10)
sp(r, t) =−
M∑
i,j=1
r,a∑
q
Kq
ij∑
l=1
⊕, ∑
p
1
2
p εqlij(σ
ql
ij )
2
∫
dvidvjdσ̂ ×
× vjiσ θ(vjiσ +
p− 1
2
vqlij )f
ij
2 (r,vi, r− qpσqlij ,vj)
qp (11)
are the flux in the local reference system and the source
which concern to the potential energy of interaction; and
ci ≡ vi −V is the thermal velocity.
So far, Eq. (1) has been being considered as the first
equation of the BBGKY hierarchy for MSP. The closure
relation for the collision integrals IE
ij , IMS
ij and the source
sp is chosen as in Ref. [19], when correlations in the
velocity space are neglected, i.e., f ij2 is replaced with
f̄ ij2 (xi, xj , t) ≡ fi(xi, t)fj(xj , t) gij2 (ri, rj , t), (12)
and gij2 is a functional of number densities nk(r, t) and
the inverse potential quasitemperature βp(r, t)
gij2 (ri, rj , t) = gij2 (ri, rj |{n}, βp), (13)
so that gij2 has the same cluster expansion (n-vertex,
f -bond) as in equilibrium. However, in the nonequilib-
rium case, nk(r, t) replaces each nk, and βp
ij(ri, rj , t) =
1
2 [βp(ri, t)+βp(rj , t)] replaces 1/kBT at each bond. The
quantity βp(r, t) is the Lagrange multiplier conjugated
to the potential energy density [19, 23–25] and is treated
in the theory using the balance equation for ep(r, t). The
functional gij2 is discontinuous at each point of disconti-
nuity of MSP and obeys the relation
gij2 (r, r± σqlij , t)
−qp = epβ
p
ijε
ql
ijgij2 (r, r± σqlij , t)
qp. (14)
Other closures are proposed for the case of a one-
component system with the square-well potential using
the equilibrium pair distribution function of the uniform
system [18] or the pair distribution function of the hard
sphere system in the state of nonuniform equilibrium [9].
The kinetic equation introduced can be used in the
high-density region only, where the contributions from
successive processes at two and more neighbouring steps
may be neglected (the pair collision approximation for
each step) [14, 15]. This approximation induces the re-
striction [15, 17]
Δσ
σ0
m
� 1
4
√
2π n (σ0
m)3 gm
2 (σ0)+
, (15)
where Δσ is the smallest separation between walls, σ0
m =
min {σ0
ij}, n is the total number density, and gm
2 (σ0)+ =
min {gij2 (σ0
ij)
+} is the smallest contact value of gij2 ’s.
On the hydrodynamic level, the system is described by
balance equations for the densities of conserved quanti-
ties, namely the mass ρ, momentum p, and energy e
densities. Such equations for ρ, p, and the kinetic en-
ergy density ek only ρ(r, t)p(r, t)
ek(r, t)
df=
M∑
i=1
〈 mi
mivi
1
2miv
2
i
〉1,i
vi
(16)
452 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
LIMITING BEHAVIOR OF THE KINETIC THEORY
can be derived from Eq. (1). In the case of M species,
they were obtained in Ref. [21]:
∂t
ρp
ek
+∇·
VρVp + P
Vek + P·V + qk
=
0
0
sk
, (17)
where V(r, t) df= p(r, t)/ρ(r, t) is the hydrodynamic ve-
locity. The stress tensor P and the heat flux qk of the
kinetic energy contain contributions of the kinetic type
(k), from repulsion at the hard core (c) due to IE
i , and
from processes at steps (MS):[
P
qk
]
=
[
Pk + Pc + PMS
qk
k + qc
k + qMS
k
]
,
where PMS = P⊗+P⊕+P , qMS
k = q⊗k +q⊕k +q k . Here,
we give only expressions for the contributions from the
processes at steps [21]:(
P
qk
)⊗
=
M∑
i,j=1
r,a∑
q
Kq
ij∑
l=1
(−q
2
)(σqlij )
3
∫
dvidvjdσ̂ vjiσ ×
× θ(vjiσ) θ(vqlij − vjiσ) σ̂
1∫
0
dλ
(
mi[c′i − ci]
1
2mi[c′2i − c2i ]
)
×
×f ij2 (r− λqσqlij ,vi, r + [1− λ]qσqlij ,vj)
−q, (18)(
P
qk
)⊕+
=
M∑
i,j=1
r,a∑
q
Kq
ij∑
l=1
⊕, ∑
p
qp
2
(σqlij )
3
∫
dvidvjdσ̂ ×
×vjiσ θ(vjiσ +
p− 1
2
vqlij ) σ̂
1∫
0
dλ
(
mi[c
qp
il − ci]
1
2mi[(c
qp
il )2 − c2i ]
)
×
×f ij2 (r + λqpσqlij ,vi, r− [1− λ]qpσqlij ,vj)
qp. (19)
The source sk on the right-hand side of Eq. (17) for ek is
such that sk = −sp identically, see Eq. (11). As a result,
the balance equation for the total energy density e = ek+
ep has no source, so that the local energy conservation
law is satisfied.
3. Kinetic Variational Theory
The main idea of KVT lies in the construction of a col-
lision integral for the potential given as a sum of the
hard-sphere repulsion and an arbitrary smooth tail
φhs+t
ij (r) = φhs
ij (r) + φt
ij(r), (20)
when a form of the functional dependence of the N -
particle distribution function is searched through maxi-
mizing the entropy subjected to certain constraints [8,9].
As a result, the pair correlations in the velocity space are
neglected, and the dependence of gij2 on {n} and βp is
determined by the type of a constraint.
In the approximation called KVT-III, the constraint
consists in the requirement that the local potential energy
density ep(r, t) is recovered correctly by the searched N -
particle distribution. The entropy maximization princi-
ple results in the function gij,hs+t
2 for φhs+t
ij in form (13).
In addition to the contribution IE
ij from φhs
ij , there is a
term of mean-field type linear in φt
ij [10]:
IE+KVT
i ≡
M∑
j=1
{
IE
ij [f
i
1, f
j
1 ] + IKVT
ij [f i1, nj ]
}
,
where IKVT
ij is given by
IKVT
ij [f i1, nj ] =
1
mi
∫
r12>σ0
ij
dr2 g
ij,hs+t
2 (r1, r2|{n}, βp)×
×nj(r2, t)[∇1φ
t
ij(r12)]· ∂1f
i
1(x1, t). (21)
The approximation accepted for gij2 is used in both IKVT
ij
and IE
ij .
For the collision integral IE+KVT
i , the authors of KVT
gave a slightly unconventional balance equation for ek in
both one- [7] and many-component [10] cases:
∂te
k+∇·
[
Vek+Pk+c·V+qk+c
k
]
+V·
(
∇·Pt
)
= Δt
ek , (22)
where the contribution to the stress tensor from the
IKVT
i reads
Pt(r1, t)
df= −1
2
M∑
i,j=1
∫
ds s
∂φt
ij(s)
∂s
×
×
1∫
0
dλnij,hs+t
2 (r1 − λs, r1 + [1− λ]s), (23)
nij,hs+t
2 (r1, r2) ≡ ni(r1, t)nj(r2, t)g
ij,hs+t
2 (r1, r2|{n}, βp).
It is not hard to recover for a general case the remainder
term Δt
ek (given in Ref. [10] in the Kac-tail limit):
Δt
ek(r1, t) =
M∑
i,j=1
[Vi −V]×
×
∫
dr21 n
ij,hs+t
2 (r1, r2|{n}, βp)φt′
ij(r12) r̂21, (24)
where Vi ≡ 〈vi〉1,ivi /ni is the average velocity of species i.
The following distinctions of Eq. (22) should be pointed
out: a) the contribution with Pt is not of the standard
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 453
Y.A. HUMENYUK, M.V. TOKARCHUK
form −∇· (Pt·V), as those with Pk or Pc; b) there is no
contribution qt
k from IKVT
i to the heat flux of the kinetic
energy; and c) due to the difference Vi −V, the source
Δt
ek is of the diffusion type and therefore vanishes in the
one-component case.
The balance equation for ep of KVT was introduced in
Ref. [10] only, where many-component system was con-
sidered, and simpler constraints which do not allow the
appearance of βp were used. It was shown in the Kac-
tail limit that the total energy density obeys a correct
balance equation through second order in gradients.
The indicated shortcomings require a more detailed
analysis of this theory.
4. Passage to the Limit in the MSP Kinetic
Theory
Starting from MSP, an arbitrary potential φ̃ij(r) can be
approximated better and better by increasing the num-
ber of steps and by a simultaneous decrease in separa-
tions between them [14]. This passage to the limit can
be denoted as
lim
{εql
ij
}→0
{Δσql
ij
}→0
(. . .) ≡ lim
φMS→φ̃
(. . .). (25)
Though the applicability condition (15) of the MSP ki-
netic theory is violated, we consider consequences of the
passage, when the model potential (20) plays the role of
the limiting potential φ̃ij(r). In this case, IE
i remains the
same, while the contribution IMS
i undergoes a change.
Transformations below resemble those applied to the
derivation [26] of the Landau kinetic equation from the
Boltzmann equation and use expansions into series in a
small value of the momentum transferred in pair colli-
sions.
4.1. The limit of the kinetic equation
Like in Ref. [14], we search for the limiting form of IMS
ij ,
Eqs. (2), (4), (5), when
f i1(ri,v
qp
il , t) f
j
1 (rj ,v
qp
jl , t) g
ij
2 (ri, rj |{n}, βp) (26)
is expanded into a series near vi and vj ; here, f̄ ij2 in-
cludes closure (12), (13).
For small step heights, {εqlij} → 0, the square root
in the pair collision rule (7) for processes ⊕ and is
expanded into a series and restricted to the linear order
in εqlij :[
vqpil
vqpjl
]
=
[
vi
vj
]
−
+
[
Mji
Mij
]
× 1
2
p
2εqlij
µijvjiσ
σ̂. (27)
For the contributions Iql⊕ij or Iql ij , this means
f̄ ij2 (r,vqpil , r + qpσqlij ,v
qp
jl )
−qp =
[
1 + p
εqlij
vjiσ
σ̂ ×
×
(
− ∂i
mi
+
∂j
mj
)
+ . . .
]
f̄ ij2 (r,vi, r + qpσqlij ,vj)
−qp, (28)
where, e.g., ∂i ≡ ∂/∂vi. The term Iql⊗ij describes the
reflection at steps of a very small height. Due to the
second θ function in Iql⊗ij , the projection vjiσ has an
upper bound 0 ≤ vjiσ ≤ (2εqlij/µij)
1/2. Therefore, v′i,v
′
j
are close to vi,vj , see Eq. (6):
f̄ ij2 (r,v′i, r− qσ
ql
ij ,v
′
j)
−q =
[
1 + 2µijvjiσσ̂ ×
×
(
− ∂i
mi
+
∂j
mj
)
+ . . .
]
f̄ ij2 (r,vi, r− qσqlij ,vj)
−q. (29)
Inserting these expressions into the formulas for Iqlpij , it
can be observed that the main contributions ∼ 1 from
processes and ⊗ in the square brackets cancel with
that from process ⊕.
The first-order term ∼vjiσ in Iql⊗ij , Eq. (29), gives the
function with a fixed upper bound being integrated over
the interval with the size tending to zero. As a result,
Iql⊗ij vanishes. The first-order terms ∼ εqlij in Eq. (28)
gives a non-zero contribution. We now consider expres-
sions from ⊕ and , e.g., for a repulsive step:
Kr
ij∑
l=1
εrlij(σ
rl
ij)
2
∫
dσ̂σ̂ ×
×
{
θ(vjiσ)
(
− ∂i
mi
+
∂j
mj
)
f̄ ij2 (r,vi, r− σrl
ij ,vj)
+−
−θ(vjiσ − vrl
ij)
(
− ∂i
mi
+
∂j
mj
)
f̄ ij2 (r,vi, r+σrl
ij ,vj)
−
}
. (30)
Some remarks concerning the passage to the limit are
as follows: i) by increasing the number of steps and de-
creasing the separations between them, the sum
∑Kq
ij
l=1 is
transformed into an integral with respect to the contin-
uous relative distance R:
Kr
ij∑
l=1
Δσr;l,l+1
ij
εrlij
Δσr;l,l+1
ij
−→
∫
φ̃′ij<0
dR
(
− ∂φ̃ij
∂R
)
, (31)
454 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
LIMITING BEHAVIOR OF THE KINETIC THEORY
where the region with φ̃′ij(R) < 0 corresponds to re-
pulsive steps and that with φ̃′ij(R) > 0 corresponds to
attractive ones; ii)
∫
dσ̂ changes to the integral over the
orientations R̂ of the relative distance vector R; iii) for
the continuous part of φ̃ij , the right and left limiting
values of f̄ ij2 are equal to each other: f̄ ij2 (.)+ = f̄ ij2 (.)−.
In those terms of the expressions obtained where f̄ ij2
depends on r − R, the integration variable should be
changed to R̂′ = −R̂. Keeping in mind that ∂φ̃ij(R)
∂R R̂ =
∂φ̃ij(R)
∂R and taking the equality of the limiting values of
f̄ ij2 into account, it can be deduced that IMS
i transforms
into
M∑
j=1
∫
dvjdR
∂φ̃ij(R)
∂R
·
(
− ∂i
mi
+
∂j
mj
)
f̄ ij2 (r,vi, r + R,vj).
(32)
The integration by parts shows that the term with ∂jf
ij
2
vanishes as f̄ ij2
∣∣
|vj |→+∞ → 0. We can integrate further
with respect to vj due to the absence of velocity corre-
lations, Eq. (26), with the result
− 1
m i
M∑
j=1
∫
dR gij2 (r, r + R|{n}, βp)×
×nj(r + R, t)
∂φ̃ij(R)
∂R
· ∂i f i1(r,vi). (33)
This expression coincides with the collision integral (21)
of the kinetic variational theory, when gij2 is allowed to
depend on βp (the version KVT-III, [8]). It is important
to stress this dependence, as the corresponding system
of limiting equations for f i1 is unclosed, until an equation
for βp(r, t) is proposed. It should be derived from the
limiting equation for ep complementing the system of
kinetic equations. The final expression (33) was obtained
for the first time in Ref. [14] for a one-component case,
but no analysis of the gij2 functional dependence was
given.
It is worth noting that Eq. (32) corresponds to the
integral term of the first equation of the BBGKY hier-
archy for a mixture with continuous potential φ̃ij . In
other words, we have shown that the pair collision oper-
ator of MSP takes, in the limit, the form of a differential
operator of pair interaction for the appropriate smooth
potential.
4.2. The limit of the equation for ep
Equation (9) is not changed, but the quantities in it are.
The quantities ep and qp contain φMS
ij and, in accordance
with Eqs. (8) and (10), go over into[
ẽp
q̃p
]
df=
M∑
i,j=1
〈1
2
[
1
ci
]
φ̃ij(rij)〉2,ijvixj
∣∣
ri→r
. (34)
The source sp depends on parameters of φMS
ij and de-
scribes the processes at steps. The contributions from ⊕
and in Eq. (11) for sp can be written explicitly as
M∑
i,j=1
r,a∑
q
Kq
ij∑
l=1
1
2
εqlij(σ
ql
ij )
2
∫
dvidvjdσ̂ vji ·σ̂ ×
×
{
θ(−vjiσ)f̄ ij2 (r,vi, r + qσqlij ,vj)
q +
+θ(vjiσ − vqlij )f̄
ij
2 (r,vi, r + qσqlij ,vj)
−q
}
, (35)
where the variable σ̂ in the term with p = ⊕ has been
changed to σ̂′ = −σ̂.
Relation (14) for the left and right limiting values of
gij2 at the discontinuity points of φMS
ij can be expanded
into a series as {εqlij} → 0,
gij2 (r, r + qσqlij , t)
−q= [1 + βp
ijε
ql
ij + . . .]gij2 (r, r + qσqlij , t)
q,
(36)
and inserted into expression (35). But the latter already
contains the factor εqlij . Therefore, only the main term in
square brackets of Eq. (36) is retained. After changing
to σ̂′ = −σ̂ in the term with q = −1, constructions (31)
can be built up with the same rules of transformation.
The two θ functions in Eq. (35) cover almost the whole
region of vjiσ with the exception of the interval of size
∼(εqlij)
1/2 which tends to zero in the limit. Replacing the
sum with the integral over the relative distance, we get
s̃p(r, t) =
M∑
i,j=1
〈1
2
vji ·
∂φ̃ij(rij)
∂rji
〉2,ijvixj
∣∣
ri→r
. (37)
This limiting expression coincides with its counterpart
for a system with smooth interaction which can be de-
duced [2], by starting immediately from the second equa-
tion of the BBGKY hierarchy and the definition of ẽp.
4.3. The limit of equations for one-particle
densities
At the passage to the limit, only the expressions for con-
tributions PMS and qMS
k are changed. The source sk
changes in the same way as sp, Eq. (37).
The contributions P⊗ and q⊗k tend rapidly to zero, as
1) due to the product θ(vjiσ) θ(v
ql
ij − vjiσ), the size of
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 455
Y.A. HUMENYUK, M.V. TOKARCHUK
the integration interval over vijσ vanishes; 2) owing to
θ(vqlij − vjiσ), the factors mi(c′i − ci) and 1
2mi[c′2i − c2i ]
vanish, as the upper bounds of their absolute values are
proportional to (εqlij)
1/2.
The contributions ⊕ and to PMS and qMS
k depend
on the height of steps through the differences which, for
small {εqlij}, are equal to Eq. (27):
[
cqpil − ci
(cqpil )2 − c2i
]
= −
[ 1
2
1
]
pMji
(vqlij )
2
cjiσ
[
σ̂
ci ·σ̂
]
.
Inserting these into Eq. (19) and going over to the con-
tinuous variable R as we have done above for IMS
ij and
sp yield[
P̃
q̃k
]⊕+
= −1
2
M∑
i,j=1
∫
dvi dvj
[
1
ci·
]∫
drji r̂jirji ×
×φ̃′ij(rij)
∫ 1
0
dλf̄ ij2 (r− λrji,vi, r + [1− λ]rji,vj). (38)
The expression for P̃⊕+ can be integrated over vi and
vj and, in the case of the potential φhs+t
ij , coincides with
the expression [7] for the contribution Pt from the soft
tail φt
ij , Eq. (23). However, the limiting heat flux of the
kinetic energy q̃⊕+
k does not have such a counterpart
in KVT.
The limiting equation for ek is of the same form as
Eq. (17), in which PMS, qMS, and sk must be replaced
with P̃⊕+ , q̃⊕+ , and s̃k = −s̃p, Eq. (37). It does not
have the shortcomings inherent in Eq. (22).
5. Conclusions
We have considered a passage to the limit of the poten-
tial of “hard spheres + arbitrary tail” for the kinetic the-
ory for the MSP system. The limiting kinetic equation
obtained coincides with that of the kinetic variational
theory [7, 9], namely KVT-III [8]. It is found out that
the contribution IMS
i from the processes at steps trans-
forms into the mean-field term of this theory. Beside the
earlier consideration for a one-component fluid [13], the
limiting form of the balance equation for the potential
energy density is additionally obtained with the explicit
expression for the corresponding source.
Contrary to the result for the kinetic equation, the lim-
iting equation for the kinetic energy density differs from
its KVT counterpart for both one- and many-component
cases. It includes additional terms with contributions to
the stress tensor and the heat flux from a smooth tail.
The reason for this disagreement will be analyzed sepa-
rately.
1. S. Chapman and T.G. Cowling, The Mathematical The-
ory of Non-Uniform Gases (Cambridge University Press,
Cambridge, 1970).
2. J.H. Ferziger and H.G. Kaper, Mathematical Theory of
Transport Processes in Gases (North-Holland, Amster-
dam, 1972).
3. H. van Beijeren and M.H. Ernst, Physica (Utrecht) 68,
437 (1973).
4. H. van Beijeren and M.H. Ernst, Physica (Utrecht) 70,
225 (1973).
5. S.A. Rice and A.R. Allnatt, J. Chem. Phys. 34, 2144
(1961).
6. R. Vogelsang and C. Hoheisel, Phys. Rev. A 39, 6391
(1989).
7. J. Karkheck and G. Stell, J. Chem. Phys. 75, 1475 (1981).
8. G. Stell, J. Karkheck, and H. van Beijeren, J. Chem.
Phys. 79, 3166 (1983).
9. J. Karkheck and G. Stell, Phys. Rev. A 25, 3302 (1982).
10. J. Karkheck, E. Martina, and G. Stell, Phys. Rev. A 25,
3328 (1982).
11. K.M. Dyer, B.M. Pettitt, and G. Stell, J. Chem. Phys.
126, 034502 (2007).
12. S. Morioka, Phys. Rev. E 72, 051203 (2005).
13. M.V. Tokarchuk and I.P. Omelyan, Preprint of Bo-
golyubov Institute for Theor. Physics, ITF–87–152R
(ITP, Kiev, 1987) (in Russian).
14. M.V. Tokarchuk and I.P. Omelyan, Preprint of Bo-
golyubov Institute for Theor. Physics, ITF–89–49R (ITP,
Kiev, 1989) (in Russian).
15. M.V. Tokarchuk and I.P. Omelyan, Ukr. Fiz. Zh. 35, 1255
(1990).
16. I.P. Omelyan and M.V. Tokarchuk, Physica A 234, 89
(1996).
17. M.V. Tokarchuk, I.P. Omelyan, and A.E. Kobryn, Phys.
Rev. E 62, 8021 (2000).
18. H.T. Davis, S.A. Rice, and J.V. Sengers, J. Chem. Phys.
35, 2210 (1961).
19. J. Karkheck, H. van Beijeren, I. de Schepper, and
G. Stell, Phys. Rev. A 32, 2517 (1985).
20. H. van Beijeren, J. Karkheck, and J.V. Sengers, Phys.
Rev. A 37, 2247 (1988).
456 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
LIMITING BEHAVIOR OF THE KINETIC THEORY
21. M.V. Tokarchuk and Y.A. Humenyuk, Condens. Matter
Phys. 10, 151 (2007).
22. J. Karkheck, G. Stell, and J. Xu, J. Chem. Phys. 89,
5829 (1988).
23. J.A. Leegwater, H. van Beijeren, and J.P.J. Michels,
J. Phys.: Condens. Matter 1, 237 (1989).
24. D.N. Zubarev and V.G. Morozov, Teor. Mat. Fiz. 60, 270
(1984).
25. J.A. Leegwater, J. Chem. Phys. 95, 8346 (1991).
26. V.P. Silin, Introduction to the Kinetic Theory of Gases
(Nauka, Moscow, 1971) (in Russian).
Received 23.10.09
ГРАНИЧНА ПОВЕДIНКА КIНЕТИЧНОЇ ТЕОРIЇ
ДЛЯ СИСТЕМ IЗ БАГАТОСХОДИНКОВИМ
ПОТЕНЦIАЛОМ
Й.А. Гуменюк, М.В. Токарчук
Р е з ю м е
Розглянуто кiнетичну теорiю сумiшей з багатосходинковим
потенцiалом взаємодiї. Проаналiзовано граничний перехiд до
плавного неперервного потенцiалу для кiнетичного рiвняння i
рiвняння балансу для густини потенцiальної енергiї. Коли гра-
ничний потенцiал вибрано у формi “твердi кульки + плавний
хвiст”, кiнетичне рiвняння зводиться до рiвняння кiнетичної
варiацiйної теорiї (КВТ), однак граничне рiвняння балансу для
густини кiнетичної енергiї вiдрiзняється вiд вiдповiдного рiв-
няння КВТ.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 457
|