On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum
Based on the generalized non-Markovian equations obtained earlier for a nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density [Markiv B.B., Omelyan I.P, Tokarchuk M.V., Condens. Matter Phys., 2010, 13, 23005] a spectrum of collective excitations is inv...
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| Cite this: | On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum / B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 14001: 1-6. — Бібліогр.: 19 назв. — англ. |
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Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. 2017-06-11T07:37:10Z 2017-06-11T07:37:10Z 2012 On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum / B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 14001: 1-6. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 82.20.M DOI:10.5488/CMP.15.14001 arXiv:1112.4743v2 https://nasplib.isofts.kiev.ua/handle/123456789/120151 Based on the generalized non-Markovian equations obtained earlier for a nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density [Markiv B.B., Omelyan I.P, Tokarchuk M.V., Condens. Matter Phys., 2010, 13, 23005] a spectrum of collective excitations is investigated, where the potential of interaction between particles is presented as a sum of the potential of hard spheres and a certain long-range potential. На основi отриманих ранiше узагальнених немаркiвських рiвнянь для нерiвноважної одночастинкової функцiї розподiлу та середнього значення густини потенцiальної частини ентальпiї [Markiv B.B., Omelyan I.P, Tokarchuk M.V., Condens. Matter Phys., 2010, 13, 23005] дослiджується спектр колективних збуджень, коли потенцiал взаємодiї мiж частинками представлено сумою потенцiалу твердих сфер та деякого далекосяжного потенцiалу. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum До проблеми узгодженого опису кiнетичних та гiдродинамiчних процесiв у густих газах та рiдинах: спектр колективних збуджень Article published earlier |
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On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum |
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On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. |
| title_short |
On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum |
| title_full |
On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum |
| title_fullStr |
On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum |
| title_full_unstemmed |
On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum |
| title_sort |
on the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: collective excitations spectrum |
| author |
Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. |
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Markiv, B.B. Omelyan, I.P. Tokarchuk, M.V. |
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2012 |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
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До проблеми узгодженого опису кiнетичних та гiдродинамiчних процесiв у густих газах та рiдинах: спектр колективних збуджень |
| description |
Based on the generalized non-Markovian equations obtained earlier for a nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density [Markiv B.B., Omelyan I.P, Tokarchuk M.V., Condens. Matter Phys., 2010, 13, 23005] a spectrum of collective excitations is investigated, where the potential of interaction between particles is presented as a sum of the potential of hard spheres and a certain long-range potential.
На основi отриманих ранiше узагальнених немаркiвських рiвнянь для нерiвноважної одночастинкової функцiї розподiлу та середнього значення густини потенцiальної частини ентальпiї [Markiv B.B., Omelyan I.P, Tokarchuk M.V., Condens. Matter Phys., 2010, 13, 23005] дослiджується спектр колективних збуджень, коли потенцiал взаємодiї мiж частинками представлено сумою потенцiалу твердих сфер та деякого далекосяжного потенцiалу.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120151 |
| citation_txt |
On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids: Collective excitations spectrum / B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 14001: 1-6. — Бібліогр.: 19 назв. — англ. |
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2025-11-26T00:18:46Z |
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2025-11-26T00:18:46Z |
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| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 1, 14001: 1–6
DOI: 10.5488/CMP.15.14001
http://www.icmp.lviv.ua/journal
Rapid Communication
On the problem of a consistent description of kinetic
and hydrodynamic processes in dense gases and
liquids: Collective excitations spectrum
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received December 21, 2011, in final form March 5, 2012
Based on the generalized non-Markovian equations obtained earlier for a nonequilibrium one-particle distri-
bution function and potential part of the averaged enthalpy density [Markiv B.B., Omelyan I.P, Tokarchuk M.V.,
Condens. Matter Phys., 2010, 13, 23005] a spectrum of collective excitations is investigated, where the potential
of interaction between particles is presented as a sum of the potential of hard spheres and a certain long-range
potential.
Key words: kinetics, hydrodynamics, kinetic equations, memory functions, collective modes
PACS: 05.20.Dd, 05.60.+w, 52.25.Fi, 82.20.M
1. Introduction
A number of investigations [1–10] were devoted to the problem of constructing a consistent descrip-
tion of kinetic and hydrodynamic processes in dense gases, liquids, and plasma. For instance, the im-
portance of taking into account the kinetic processes connected with irreversible collision processes at
the scale of short-ranged interparticle interactions was pointed out in [11]. Short-wavelength collective
modes in liquids were investigated therein based on the linearized kinetic equation of the revised Enskog
theory for the hard spheres model.
In this paper we investigate a spectrum of collective excitations within a consistent description of
kinetic and hydrodynamic processes in a system in which the potential of interaction between particles
consists of two parts: the hard spheres potential and a long-range part.
2. Transport equations
Using the ideas presented in papers [3, 4] the nonequilibrium statistical operator consistently describ-
ing the kinetic and hydrodynamic processes for a system of classical interacting particles was obtained
in [7, 8] by means of nonequilibrium statistical operator method. Using this operator, a set of kinetic
equations for the nonequilibrium one-particle distribution function f~k (~p; t) = 〈n̂~k
(~p)〉t and the potential
part of the averaged enthalpy density hint
~k
(t) = 〈ĥint
~k
〉t was obtained in the case of weakly nonequilibrium
processes:
∂
∂t
f~k (~p; t)+
i~k ·~p
m
f~k (~p; t) =−
i~k ·~p
m
n f0(p)c2(k)
∫
d~p ′ f~k (~p ′
; t)+ iΩnh (~k;~p)hint
~k
(t)
−
∫
d~p ′
t
∫
−∞
e
ε(t ′−t )ϕnn(~k;~p,~p ′
; t , t ′) f~k (~p ′
; t ′)dt ′−
t
∫
−∞
e
ε(t ′−t )ϕnh(~k ;~p; t , t ′)hint
~k
(t ′)dt ′, (1)
© B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk, 2012 14001-1
http://dx.doi.org/10.5488/CMP.15.14001
http://www.icmp.lviv.ua/journal
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
∂
∂t
hint
~k
(t) =
∫
d~p ′
iΩhn (~k;~p ′
) f~k (~p ′
; t)
−
∫
d~p ′
t
∫
−∞
e
ε(t ′−t )ϕhn(~k;~p ′
; t , t ′) f~k (~p ′
; t ′)dt ′−
t
∫
−∞
e
ε(t ′−t )ϕhh(~k ; t , t ′)hint
~k
(t ′)dt ′, (2)
where iΩnh (~k;~p) = 〈 ˙̂n~k
(~p)ĥint
−~k
〉0Φ
−1
hh
(~k) and iΩhn (~k;~p) =
∫
d~p ′〈 ˙̂
hint
~k
n̂−~k (~p ′)〉0Φ
−1
~k
(~p ′,~p) are the normal-
ized static correlation functions.
ϕnn(~k;~p,~p ′
; t , t ′) =
∫
d~p ′′〈In (~k;~p)T0(t , t ′)In (−~k;~p ′′
)〉0Φ
−1
~k
(~p ′′
,~p ′
),
ϕhn(~k;~p; t , t ′) =
∫
d~p ′〈I int
h (~k)T0(t , t ′)In (−~k;~p ′
)〉0Φ
−1
~k
(~p ′
,~p),
ϕnh(~k;~p; t , t ′) = 〈In (~k;~p)T0(t , t ′)I int
h (−~k)〉0Φ
−1
hh(~k),
ϕhh (~k; t , t ′) = 〈I int
h (~k)T0(t , t ′)I int
h (−~k)〉0Φ
−1
hh (~k) (3)
are the generalized transport kernels (memory functions) describing kinetic and hydrodynamic pro-
cesses. In [12], the inner structure of generalized transport kernels for a consistent description of ki-
netic and hydrodynamic processes was analyzed in detail. It was shown that they are expressed in
terms of time correlation functions related to the basic set of dynamical variables, phase density n̂~k
(~p)
and potential part of the enthalpy density ĥint
~k
along with the transport kernels describing diffusive
and visco-thermal processes. Here, n̂~k
(~p) =
∫
d~r e−i~k~r n̂1(~r ,~p) are the Fourier-components of microscopic
phase density of particles number, n̂1(~r ,~p) =
∑N
l=1
δ(~p −~pl )δ(~r −~rl ), ĥint
~k
= ε̂int
~k
−〈ε̂int
~k
n̂−~k 〉0S−1(k)n̂~k
are
the Fourier-components of the potential part of the enthalpy density, ε̂int
~k
= 1
2
∑N
l, j=1
Φ(|~rl j |)e−i~k~r j and
n̂~k
=
∑N
l=1
e−i~k~rl are the Fourier-components of the potential energy and particle number densities, re-
spectively, ~k is the wave-vector. Φ−1
hh
(~k) is the function inverse to the equilibrium correlation function
Φhh(~k) = 〈ĥint
~k
ĥint
−~k
〉0, 〈. . .〉0 =
∫
dΓN . . .̺0(xN ), where ̺0 is an equilibrium statistical operator. In (~k;~p) =
(1−P0)iLN n̂~k
(~p) = (1−P0) ˙̂n~k
(~p) and I int
h
(~k) = (1−P0)iLN ĥint
~k
= (1−P0)
˙̂
hint
~k
are the generalized flows
in linear approximation, T0(t , t ′) = e(t−t ′)(1−P0)iLN is the evolution operator with regard to projection.
P0 is the linear approximation of the Mori projection operator constructed on the orthogonal dynamic
variables n̂~k
(~p), ĥint
~k
[8]: P0 Â~k
=
∑
~k
〈Â~k
ĥint
−~k
〉0Φ
−1
hh
(~k)ĥint
~k
+
∑
~k
∫
d~p
∫
d~p ′〈Â~k
n̂−~k (~p)〉0Φ
−1
~k
(~p,~p ′)n̂~k
(~p ′). It
possesses the following properties: P0P0 = P0, P0(1−P0) = 0, P0n̂~k
(~p) = n̂~k
(~p), P0ĥint
~k
= ĥint
~k
. Φ−1
~k
(~p,~p ′)
is the function inverse to Φ~k
(~p,~p ′) = 〈n̂~k
(~p)n̂−~k (~p ′)〉0 = nδ(~p −~p ′) f0(p ′)+n2 f0(p) f0(p ′)h2(~k). It is equal
to Φ
−1
~k
(~p,~p ′) = δ(~p−~p′)
n f0(p′) −c2(k), where n = N /V , f0(p)=
(
β/2πm
)3/2
e−β
p2
2m is the Maxwellian distribution,
β = 1/kBT is an inverse temperature and kB is Boltzmann constant. c2(k) is the direct correlation func-
tion related to the correlation function h2(k): h2(k) = c2(k)[1−nc2(k)]−1. S(k) = 〈n̂~k
n̂−~k 〉0 denotes the
static structure factor. It is important to note that dynamical variables ĥint
~k
and n̂~k
(~p) are orthogonal in
the sense that 〈ĥint
~k
n̂~k
(~p)〉0 = 0.
Projecting the set of equations (1), (2) onto the first moments of the nonequilibrium one-particle dis-
tribution function Ψ1(~p) = 1, Ψα(~p) =
p
2pα/2kBT (where α= x, y, z), Ψε(~p) =
p
2/3(p2/2mkBT −3/2),
one can obtain a set of equations for the averaged values of densities of particles number n~k
(t), momen-
tum ~~k (t), kinetic hkin
~k
(t) and potential hint
~k
(t) parts of enthalpy [8], where the Fourier-components of
the kinetic part of enthalpy density defined as ĥkin
~k
= ε̂kin
~k
−〈ε̂kin
~k
n̂−~k〉0〈n̂~k
n̂−~k〉
−1
0
n̂~k
. For this purpose, we
introduce the projection operator constructed on the eigenfunctions |Ψν(~p)〉 of the nonequilibrium one-
particle function such that P |Ψ〉 =
∑n
ν=1
|Ψν〉〈Ψν|Ψ〉. Here, 〈Ψ|Ψν〉 =
∫
d~pΨ(~p) f0(p)Ψν(~p), while Ψν(~p)
satisfies the conditions 〈Ψµ|Ψν〉 = δµν and
∑
ν |Ψν〉〈Ψν| = 1. Then, let us act by the projection operator P
onto the set of equation (1), (2). Repeat this operation acting by the operator Q = 1−P complementary
to P . Then, substituting the unknown quantity from the second equation into the first one we obtain
the necessary set of equations with separated contributions of kinetic and potential energies. Using the
14001-2
On the problem of a consistent description...
Laplace transform, let us represent it in a matrix form:
zã~k (z)− Σ̃G(~k ; z)ã~k (z)=−〈ã~k (t = 0)〉t
. (4)
Σ̃G(~k; z) is the matrix of memory kernels
Σ̃G(~k; z) = iΩ̃G(~k)− Π̃(~k; z), (5)
where ã~k (z) = [n~k
(z),~~k (z),hkin
~k
(z),hint
~k
(z)] is the column-vector.
iΩ̃G(~k) =
0 iΩn 0 0
iΩn 0 iΩkin
h
iΩint
h
0 iΩkin
h
0 0
0 iΩint
h
0 0
, Π̃(~k; z) =
0 0 0 0
0 Π Π
kin
h
Π
int
h
0 Π
kin
h
Π
kin,kin
hh
Π
kin,int
hh
0 Π
int
h
Π
int,kin
hh
Π
int,int
hh
(6)
are the frequency matrix and the matrix of transport kernels. The elements of the latter have the follow-
ing structure:
Πµν(~k; z) = 〈Ψµ|ϕ̃(~k; z)+ Σ̃(~k; z)Q[zĨ −QΣ̃(~k; z)Q]
−1
QΣ̃(~k; z)|Ψν〉. (7)
Ĩ denotes a unit matrix, ϕ̃(~k; z) is the matrix whose elements are the generalized transport kernels
ϕnn(~k;~p,~p ′; z), ϕhn(~k;~p ′; z), ϕnh (~k;~p; z), ϕhh(~k; z) in the set of equations (1), (2), and Σ̃(~k; z) = iΩ̃(~k)−
ϕ̃(~k; z). Here, iΩ̃(~k) is the matrix of static correlation functions iΩnh (~k;~p), iΩhn (~k;~p). For the sake of
simplicity the dependence of ϕ̃(~k; z), Σ̃(~k; z) on ~p,~p ′ was omitted. As we can see from the structure of ele-
ments of the matrices iΩ̃G(~k) and Π̃( ~k; z), the contributions of kinetic and potential parts of enthalpy are
separated. Herewith, a question arises regarding the study of time correlation functions and collective
modes for liquids based on the set of transport equations (4).
3. Spectrum of collective excitations
Let us consider the system of kinetic equations (1), (2) in the case where the potential of interaction is
presented as follows:
Φ(|~ri j |) =Φ
hs
(|~ri j |)+Φ
l
(|~ri j |), (8)
where Φ
hs(|~ri j |) is the hard sphere interaction potential, and Φ
l(|~ri j |) is the long-range potential. Taking
into account the features of the hard sphere model dynamics [4] and the results of investigations [11, 13,
14], one can separate Enskog-Boltzmann collision integral from the function ϕnn(~k ;~p,~p ′; t , t ′). Indeed, an
infinitesimal time of a collision τ0 →+0 within an infinitesimal regionσ±∆r0,∆r0 ∼ |τ0||~p2−~p1|/m →+0
being a feature of the hard sphere model dynamics (σ is the hard sphere diameter). Taking this into ac-
count in the kinetic equation (1) we can obtain the kinetic equation of the revised Enskog theory for
the hard sphere model and the kinetic Enskog-Landau equation for the charged hard sphere model in a
pair collision approximation, respectively [4]. In the latter case, when Φ
l(|~ri j |) is the Coulomb potential
of interaction, taking into account the features τ0 →+0, ∆r0 →+0 makes it possible to separate a colli-
sion integral of the revised Enskog theory and a Landau-like collision integral in the limits τ→−0 and
τ → −∞, respectively. In the case of potential (8), in the region of τ0 → +0, ∆r0 → +0, σ±∆r0 where
the main contribution to a dynamics is defined by pair collisions of hard spheres, the memory function
ϕnn(~k;~p,~p ′; t , t ′) can be calculated by expanding it over the density (a pair collision approximation),
which was scrupulously done in papers by Mazenko [13–17].
Then, the kinetic equation (1) can be represented in the following form:
∂
∂t
f~k (~p; t)+
i~k ·~p
m
f~k (~p; t) =−
i~k ·~p
m
n f0(~p)
[
c2(k)− g2(σ)c0
2 (k)
]
∫
d~p ′ f~k (~p ′
; t)
−
∫
d~p ′ϕhs
nn(~k ,~p,~p ′
) f~k (~p ′
; t)+ iΩnh (~k;~p)hint
~k
(t)
−
∫
d~p ′
t
∫
−∞
dt ′eε(t−t ′)ϕl
nn(~k ;~p,~p ′
; t , t ′) f~k (~p ′
; t ′)−
t
∫
−∞
dt ′eε(t−t ′)ϕnh(~k;~p; t , t ′)hint
~k
(t ′). (9)
14001-3
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
Here,
∫
d~p ′ϕhs
nn(~k,~p,~p ′
) f~k (~p ′
; t) = ng2(σ)σ2
∫
dΩσ
∫
d~p ′ (~p −~p ′) ·~̂σ
m
Θ−
(
~̂σ · [~p −~p ′
]
)
×
[
f0(p ′∗
) f~k (~p; t)− f0(p ′
) f~k (~p∗
; t)+e
i~k·~̂σσ f0(p ′∗
) f~k (~p ′∗
; t)−e
i~k·~̂σσ f0(p) f~k (~p ′
; t)
]
(10)
is the Enskog-Boltzmann collision integral, where c0
2
(~k) is the low-density limit of the direct correlation
function and g2(σ) is the pair distribution function. The step function Θ−(x) is unity for x < 0 and van-
ishes otherwise. dΩσ is the differential solid angle, ~̂σ is unity vector. The precollision and postcollision
momenta of the colliding hard spheres are denoted as (~p,~p ′) and (~p∗,~p ′∗), respectively. ϕl
nn(~k;~p,~p ′; t , t ′)
is the part of the transport kernel related to the long-range interaction potential Φl(|~ri j |). Notably, the
presented equation contains the Enskog-Boltzmann collision integral describing short-time dynamics of
the hard sphere model. The collective effects related to the long-range interactions between particles are
described by the functions iΩnh (~k ;~p), ϕl
nn(~k;~p,~p ′; t , t ′),ϕnh (~k; t , t ′) and by the equation for hint
~k
(t). Since
the collective modes for the Enskog-Boltzmann model are well studied [11], the investigation of time cor-
relation functions and collective modes for the system of particles interacting through the potential (8)
turns out to be of great interest. In the case of the hard spheres system, the set of kinetic equations (2), (9)
reduces to the Enskog-Boltzmann kinetic equation [11].
∂
∂t
f~k (~p; t)+
i~k ·~p
m
f~k (~p; t) =−
i~k ·~p
m
n f0(~p)
[
c2(k)− g2(σ)c0
2(k)
]
∫
d~p ′ f~k (~p ′
; t)
−ng2(σ)σ2
∫
dΩσ
∫
d~p ′ (~p −~p ′) ·~̂σ
m
Θ−
(
~̂σ · [~p −~p ′
]
)
×
[
f0(p ′∗
) f~k (~p; t)− f0(p ′
) f~k (~p∗
; t)+e
i~k ·~̂σσ f0(p ′∗
) f~k (~p ′∗
; t)−e
i~k ·~̂σσ f0(p) f~k (~p ′
; t)
]
. (11)
Projecting the Enskog-Boltzmann equation (11) onto the first moments of the nonequilibrium one-particle
distribution function a spectrum of collective excitations for the hard sphere model was obtained in [11,
18]. Herewith, it is important to note that for the kinetic Enskog-Boltzmann equation we can consider two
typical limits: kσ≪ 1 and kσ≫ 1. In the hydrodynamic limit (kσ≪ 1) the spectrum includes: heatmode
zH(k) =−DTEk2, where DTE is the thermal diffusivity coefficient in the Enskog transport theory [19]; two
soundmodeswith eigenvalues given by z±(k) =±ick −ΓEk2, where ΓE is the sound damping coefficient
and c is the sound velocity in the Enskog theory; two shear modes with eigenvalues given by zν1
(k) =
zν2
(k) = zν(k) =−νEk2, νE is the kinematic viscosity in the Enskog dense gas theory. In the limit kσ≫ 1
the Enskog-Boltzmann collision integral (10) is transformed [11] into the Lorentz-Boltzmann collision
integral which has only one eigenfunction Ψ1(~p) = 1. Consequently, we obtain the diffusion mode only
with the eigenvalue zD(k) = −DEk2, where DE is the self-diffusion coefficient as given by the Enskog
dense gas theory.
Let us now project the system of equations (2), (9) onto the first moments of the nonequilibrium
one-particle distribution function. Thereafter, we perform simple transformations consisting in the tran-
sition from the set of equations (4) for averages n~k
(z),~~k (z),hkin
~k
(z),hint
~k
(z) to the equations of general-
ized hydrodynamics for averages b̃~k (z) = [n~k
(z),~~k (z),h~k
(z)= hkin
~k
(z)+hint
~k
(z)]. This permits to correctly
define (see below) the generalized viscosity coefficient via the transport kernel (13) and the heat con-
ductivity coefficient via the transport kernel Πhh (k, z). The averages b̃~k (z) satisfy the set of equations
zb̃~k (z) − Σ̃G(~k; z)b̃~k (z) = −〈b̃~k (t = 0)〉t . In the limit kσ ≫ 1, the latter reduces to a single equation of
diffusion for n~k
(z) in which the transport kernel ΣG(~k; z) = 〈Ψ1|ϕL−B
nn (~k)|Ψ1〉 corresponds to the Lorentz-
Boltzmann collision integral (10). In the opposite case, when kσ ≪ 1, the matrix Σ̃G(~k; z) is defined as
follows: Σ̃G(~k; z) = Σ̃H(~k; z) = iΩ̃H(~k)− Π̃H(~k; z),
Σ̃H(~k; z) =
0 iΩn 0
iΩn −〈Ψ2|ϕhs
nn |Ψ2〉−Σ
l
j j
iΩh −Π h
0 iΩh −Πh −〈Ψ3|ϕhs
nn |Ψ3〉−Π
l
hh
(k ,z)
. (12)
14001-4
On the problem of a consistent description...
Here we use the notations
Σ j j (k, z) = Π j j (k, z)−Σ
int
j h (k, z)
[
Σ
kin,kin
hh
(k, z)
]−1 {
iΩ
kin
h j (k)+Π
kin,int
hh
(k, z) (13)
×
[
z −Σ
int,int
hh
(k, z)
]−1
Σ
int
h j (k, z)
}
−Σ
kin
j h (k, z)
[
z −Σ
int,int
hh
(k, z)
]−1
Σ
int
h j (k, z),
Πhh (k, z) = Π
kin,kin
hh
(k, z)+Π
kin,int
hh
(k, z)+Π
int,kin
hh
(k, z)+Π
int,int
hh
(k, z), (14)
where Σ
kin,kin
hh
(k, z) = z − Π
kin,kin
hh
(k, z), Σ
int,int
hh
(k, z) = Π
int,int
hh
(k, z) + Π
int,kin
hh
(k, z)
[
Σ
kin,kin
hh
(k, z)
]−1
×Πkin,int
hh
(k, z), Σint
h j
(k, z) = iΩint
h j
(k)+Πint,kin
hh
(k, z)
[
Σ
kin,kin
hh
(k, z)
]−1
iΩkin
h j
(k). We can separate real and imag-
inary parts in memory functions (13) and (14) as follows: Σ j j (k, z) =Σ
′
j j
(k,ω)+iΣ′′
j j
(k,ω) andΠhh (k, z) =
Π
′
hh
(k,ω)+ iΠ′′
hh
(k,ω). Herewith, the contributions from the hard sphere dynamics with typical spatial-
temporal scale τ0 →+0, ∆r0 →+0 are separated in the transport kernel ϕnn(~k;~p,~p ′; t , t ′) only in the first
term in the right-hand side of elements (7) and hence in (13). After these transformations we can ob-
tain a spectrum of collective excitations in the hydrodynamic limit kσ≪ 1: heat mode zH(k) =−DTk2,
where DT is the thermal diffusivity coefficient for the system with the potential of interaction (8). It
has the following structure: DT = DTE +D l
T
, D l
T
is determined through the corresponding elements (7)
of matrix of transport kernels (6). D l
T
= λl
nmcp
, cp is a heat capacity at constant pressure, λl is the heat
conductivity coefficient in the hydrodynamic limit: λl = limk→0,ω→0λ
l(k,ω). λl(k,ω) is the generalized
heat conductivity coefficient defined via elements of the matrix (6): λl(k,ω) = cV (k)
kBβ
2
1
k2 Π
′′
hh
(k,ω), where
cV (k) is the generalized heat capacity at constant volume dependent on the wave vector ~k ; two sound
modes z±(k) = ± ick − Γk2, where Γ is the sound damping and c = cp
cV βmS(0)
is the sound velocity in
the system with the potential of interaction (8), S(0) = S(k = 0), S(k) is a static structure factor of the
system with potential (8). Γ = 1
2
(cp /cV −1)DT + 1
2
ηL, where cV = cV (k = 0), ηL =
(
4
3
η⊥+ηb
)
/mn is the
longitudinal viscosity defined via the bulk viscosity ηb = ηb
E
+ηb
l
and the shear viscosity η⊥ = η⊥
E
+η⊥
l
coefficients. η⊥
E
is the shear viscosity in Enskog theory, and η⊥
l
is calculated in the hydrodynamic limit
η⊥
l
= limk→0,ω→0η
⊥
l
(k,ω). η⊥
l
(k,ω) is the generalized shear viscosity coefficient defined via elements
of the matrix (6) η⊥
l
(k,ω) = mn
β
1
k2 Σ
′′⊥
j j
(k,ω). Σ⊥
j j
(k, z) is the transverse component of the generalized
transport kernel Σ j j (k, z), where the wave vector ~k is directed along the 0Z axis. The longitudinal vis-
cosity coefficient ηL
l
is calculated in the hydrodynamic limit ηL
l
= limk→0,ω→0η
L
l
(k,ω), where ηL
l
(k,ω) is
the generalized longitudinal viscosity coefficient defined via longitudinal components of the generalized
transport kernel Σ j j (k, z): ηL
l
(k,ω) = mn
β
1
k2 Σ
′′L
j j
(k,ω); two shear modes with the eigenvalues given by
zν(k) =−νk2. ν= νE +νl is the kinematic viscosity ν= η⊥/nm for the system with the potential of inter-
action (8). Here, νl is a contribution determined by the corresponding elements (7) of the matrix of trans-
port kernels (6). In the limit kσ≫ 1, we obtain a diffusion mode, with the eigenvalue zD(k) = −DEk2,
which is the same as in the Enskog theory.
As we can see from the above expressions, presence of the long-range part in the potential of in-
teraction entails a renormalization of all the damping coefficients in the collective modes spectrum. In
particular, contributions related to long-range potential appear in heat and sound modes as well as in
shear modes. Nevertheless, diffusion mode remains unchanged.
4. Conclusions
In this brief report within the framework of consistent description of kinetic and hydrodynamic pro-
cesses we considered a set of kinetic equations for the potential of interaction of the system presented by
the sum of hard spheres potential Φhs(|~ri j |) and a certain smooth one Φl(|~ri j |). In this case, we separated
the Enskog-Boltzmann collision integral describing a collision dynamics at short distances from the colli-
sion integral of the kinetic equation for the nonequilibrium distribution function. Applying the procedure
of projecting onto the moments of the nonequilibrium distribution function to the equations (2), (9) we
obtain a set of equations for hydrodynamic variables. Based on this set of equations a spectrum of collec-
tive excitations was obtained in the limits kσ≪ 1 and kσ≫ 1. We showed that, besides the contribution
14001-5
B.B. Markiv, I.P. Omelyan, M.V. Tokarchuk
from the hard spheres potential, all hydrodynamic modes contain contributions from the long-range part
of potential. These contributions make the damping coefficients closer to the ones known from the hy-
drodynamic theory. Here, we formally presented the contribution from the long-ranged part of potential,
since the latter, for example the Coulomb one, will contribute into the transport kernels (3). Moreover, we
can separate the linearized Landau-like collision integral describing pair collisions in ϕnn(~k;~p,~p ′; t , t ′),
while ϕhn(~k;~p; t , t ′), ϕnh (~k;~p; t , t ′), ϕhh (~k;~p; t , t ′) take into account collective Coulombic interactions.
Evidently, calculation of the elements (7) of matrix Π̃(~k ; z) will depend on the model of time dependence
(exponential, Gaussian etc.) for transport kernels (3). When a spectrum of collective excitations is known,
awhole set of time correlation functions can be investigated. In particular, it makes possible to investigate
the behaviour of the dynamic structure factor and, in the case of potential (8), to separate a contributions
from the hard spheres potential and the long-range part of potential in it.
References
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1970.
До проблеми узгодженого опису кiнетичних та
гiдродинамiчних процесiв у густих газах та рiдинах:
спектр колективних збуджень
Б.Б. Маркiв, I.П. Омелян, М.В. Токарчук
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
На основi отриманих ранiше узагальнених немаркiвських рiвнянь для нерiвноважної одночастинко-
вої функцiї розподiлу та середнього значення густини потенцiальної частини ентальпiї [Markiv B.B.,
Omelyan I.P, Tokarchuk M.V., Condens. Matter Phys., 2010, 13, 23005] дослiджується спектр колективних збу-
джень, коли потенцiал взаємодiї мiж частинками представлено сумою потенцiалу твердих сфер та деякого
далекосяжного потенцiалу.
Ключовi слова: кiнетика, гiдродинамiка, кiнетичнi рiвняння, функцiї пам’ятi, колективнi моди
14001-6
http://dx.doi.org/10.1103/PhysRevA.12.254
http://dx.doi.org/10.1007/BF01018982
http://dx.doi.org/10.1007/BF01016582
http://dx.doi.org/10.1007/BF01015557
http://dx.doi.org/10.1007/BF01014805
http://dx.doi.org/10.1007/BF01019063
http://dx.doi.org/10.1007/BF01008939
http://dx.doi.org/10.5488/CMP.13.23005
http://dx.doi.org/10.1103/PhysRevA.12.1653
http://dx.doi.org/10.1103/PhysRevA.9.360
http://dx.doi.org/10.1103/PhysRevA.3.2121
http://dx.doi.org/10.1103/PhysRevA.5.2545
http://dx.doi.org/10.1103/PhysRevA.6.1981
http://dx.doi.org/10.1103/PhysRevA.22.287
Introduction
Transport equations
Spectrum of collective excitations
Conclusions
|