Dielectric relaxation in dipolar fluids. Generalized mode approach
The concept of generalized collective modes, recently proposed for the investigation of simple fluids, is now applied to describe processes of dielectric relaxation in dipolar systems. The approach presented here is an extension of the dipole-density formalism to arbitrary numbers of dynamical v...
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Інститут фізики конденсованих систем НАН України
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| Cite this: | Dielectric relaxation in dipolar fluids. Generalized mode approach / I.P. Omelyan, I.M. Mryglod, M.V. Tokarchuk // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 179-200. — Бібліогр.: 30 назв. — англ. |
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Omelyan, I.P. Mryglod, I.M. Tokarchuk, M.V. 2017-05-31T18:24:45Z 2017-05-31T18:24:45Z 1998 Dielectric relaxation in dipolar fluids. Generalized mode approach / I.P. Omelyan, I.M. Mryglod, M.V. Tokarchuk // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 179-200. — Бібліогр.: 30 назв. — англ. 1607-324X DOI:10.5488/CMP.1.1.179 PACS: 67.55.Jd; 77.22.-d; 47.65 https://nasplib.isofts.kiev.ua/handle/123456789/118824 The concept of generalized collective modes, recently proposed for the investigation of simple fluids, is now applied to describe processes of dielectric relaxation in dipolar systems. The approach presented here is an extension of the dipole-density formalism to arbitrary numbers of dynamical variables and values of wavelengths. Generalized dipolar mode spectra of a Stockmayer fluid are evaluated over a wide scale of wavelengths up to the five-variable approximation. The wavevector- and frequency-dependent dielectric permittivity and dipole-moment time autocorrelation functions are calculated on the basis of analytical expressions using the dipolar modes. The obtained results are compared with those achieved in lower-order approximations and with molecular dynamics data. It is shown that the fivevariable description quantitatively reproduces the entire frequency dependence of the dielectric constant at arbitrary wavenumbers. Концепцiя узагальнених колективних мод, яка недавно була запpопонована для дослiдження пpостих piдин, застосовується заpаз до опису пpоцесiв дiелектpичної pелаксацiї в дипольних системах. Даний пiдхiд пpедставляє собою pозшиpення фоpмалiзму дипольної густини до довiльного числа динамiчних змiнних i значень довжин хвиль. Спектpи узагальнених дипольних мод Штокмайєpiвської piдини визначено в шиpокiй областi довжин хвиль у наближенняx аж до п’ятого поpядку. Дiелектpична спpийнятливiсть, залежна вiд хвильового вектоpа i частоти, та часова автокоpеляцiйна функцiя дипольного моменту обчислюються на основi аналiтичних виpазiв, викоpистовуючи дипольнi моди. Отpиманi pезультати поpiвнюються з pезультатами нижчепоpядкових наближень i даними молекуляpної динамiки. Показано, що п’яти-модовий опис кiлькiсно вiдтвоpює дiелектpичну функцiю в усiй областi змiни хвильового вектоpа i частоти. One of us (I.P.) would like to acknowledge financial support of the President of Ukraine. I.M. also thanks the “Fonds für Förderung der wissenschaftlichen Forschung” under Project P12422TPH for financial support. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Dielectric relaxation in dipolar fluids. Generalized mode approach Діелектрична релаксація у дипольних рідинах. Підхід узагальнених мод Article published earlier |
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| title |
Dielectric relaxation in dipolar fluids. Generalized mode approach |
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Dielectric relaxation in dipolar fluids. Generalized mode approach Omelyan, I.P. Mryglod, I.M. Tokarchuk, M.V. |
| title_short |
Dielectric relaxation in dipolar fluids. Generalized mode approach |
| title_full |
Dielectric relaxation in dipolar fluids. Generalized mode approach |
| title_fullStr |
Dielectric relaxation in dipolar fluids. Generalized mode approach |
| title_full_unstemmed |
Dielectric relaxation in dipolar fluids. Generalized mode approach |
| title_sort |
dielectric relaxation in dipolar fluids. generalized mode approach |
| author |
Omelyan, I.P. Mryglod, I.M. Tokarchuk, M.V. |
| author_facet |
Omelyan, I.P. Mryglod, I.M. Tokarchuk, M.V. |
| publishDate |
1998 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Діелектрична релаксація у дипольних рідинах. Підхід узагальнених мод |
| description |
The concept of generalized collective modes, recently proposed for the
investigation of simple fluids, is now applied to describe processes of dielectric
relaxation in dipolar systems. The approach presented here is an
extension of the dipole-density formalism to arbitrary numbers of dynamical
variables and values of wavelengths. Generalized dipolar mode spectra
of a Stockmayer fluid are evaluated over a wide scale of wavelengths up to
the five-variable approximation. The wavevector- and frequency-dependent
dielectric permittivity and dipole-moment time autocorrelation functions are
calculated on the basis of analytical expressions using the dipolar modes.
The obtained results are compared with those achieved in lower-order approximations
and with molecular dynamics data. It is shown that the fivevariable
description quantitatively reproduces the entire frequency dependence
of the dielectric constant at arbitrary wavenumbers.
Концепцiя узагальнених колективних мод, яка недавно була запpопонована для дослiдження пpостих piдин, застосовується заpаз до опису пpоцесiв дiелектpичної pелаксацiї в дипольних системах. Даний пiдхiд пpедставляє собою pозшиpення фоpмалiзму дипольної густини до довiльного числа динамiчних змiнних i значень довжин хвиль. Спектpи узагальнених дипольних мод Штокмайєpiвської piдини визначено в шиpокiй областi довжин хвиль у наближенняx аж до п’ятого поpядку. Дiелектpична спpийнятливiсть, залежна вiд хвильового вектоpа i частоти, та часова автокоpеляцiйна функцiя дипольного моменту обчислюються на основi аналiтичних виpазiв, викоpистовуючи дипольнi моди. Отpиманi pезультати поpiвнюються з pезультатами нижчепоpядкових наближень i даними молекуляpної динамiки. Показано, що п’яти-модовий опис кiлькiсно вiдтвоpює дiелектpичну функцiю в усiй областi змiни хвильового вектоpа i частоти.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118824 |
| citation_txt |
Dielectric relaxation in dipolar fluids. Generalized mode approach / I.P. Omelyan, I.M. Mryglod, M.V. Tokarchuk // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 179-200. — Бібліогр.: 30 назв. — англ. |
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2025-11-25T04:25:08Z |
| last_indexed |
2025-11-25T04:25:08Z |
| _version_ |
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| fulltext |
Condensed Matter Physics, 1998, Vol. 1, No 1(13), p. 179–200
Dielectric relaxation in dipolar fluids.
Generalized mode approach
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
Institute for Condensed Matter Physics of the National Academy of
Sciences of Ukraine, 1 Svientsitskii St., UA–290011 Lviv–11, Ukraine
Received November 28, 1997
The concept of generalized collective modes, recently proposed for the
investigation of simple fluids, is now applied to describe processes of di-
electric relaxation in dipolar systems. The approach presented here is an
extension of the dipole-density formalism to arbitrary numbers of dynami-
cal variables and values of wavelengths. Generalized dipolar mode spectra
of a Stockmayer fluid are evaluated over a wide scale of wavelengths up to
the five-variable approximation. The wavevector- and frequency-dependent
dielectric permittivity and dipole-moment time autocorrelation functions are
calculated on the basis of analytical expressions using the dipolar modes.
The obtained results are compared with those achieved in lower-order ap-
proximations and with molecular dynamics data. It is shown that the five-
variable description quantitatively reproduces the entire frequency depen-
dence of the dielectric constant at arbitrary wavenumbers.
Key words: collective modes; dielectric properties; extended
hydrodynamics
PACS: 67.55.Jd; 77.22.-d; 47.65
1. Introduction
Processes of dielectric relaxation in polar fluids have been intensively studied
in theory [1–7], computer simulation [8–15] and pure experiment [16–18]. Never-
theless, there is still lack of agreement between the theoretically predicted and
experimental results. Moreover, some main problems remain unsolved even in the
simplest case when the fluid is treated as a system of point dipoles. Among various
theoretical schemes, enabling us to describe the processes of dielectric relaxation in
dipolar fluids, it is necessary to point out two consequent approaches: the dipole-
density formalism of Madden and Kivelson [2] and the extended hydrodynamic
description popularized by Bagchi and Chandra [4].
The usual Navier-Stokes hydrodynamics [19], being valid in the infinite-wave-
length limit and large times, cannot be used at molecular length scales. In the ex-
tended hydrodynamic description, the microscopic operators of the number density
c© I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk 179
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
n̂(r,Θ) =
∑N
i=1 δ(r−ri)δ(Θ−Θi), spatial P̂ (r,Θ) =
∑N
i=1mviδ(r−ri)δ(Θ−Θi)
and angular L̂(r,Θ) =
∑N
i=1 Jwiδ(r−ri)δ(Θ−Θi) momenta densities are consid-
ered as basic dynamical quantities [6]. These quantities satisfy modified equations
which take into account processes with short and intermediate time scales as well.
In their theory, fluctuations of the total energy density ê(r,Θ) =
∑N
i=1 eiδ(r −
ri)δ(Θ−Θi), where ei =
1
2
(
mv2
i + Jw2
i +
∑N
j(j 6=i) ϕij
)
, are not taken into account
because of the complexities of hydrodynamic equations. Vector Θi characterizes
orientations of molecule i and, in the case of rigid nonpolarizable molecules, it can
be associated with the unit vector directed along the particle’s dipole moment,
i.e., Θi = µi/µ. Then, the dipole-moment fluctuations can be reproduced on the
basis of number-density correlations using the relation M̂(r, t) = µ〈Θ n̂(r,Θ, t)〉
Θ
where the averaging is performed over orientations. In such a way the dynamical
polarization in a dense dipolar fluid can be investigated, provided reasonable forms
of the dissipative kernels are available. Despite the fact that such an approach al-
lows us, in principle, to describe the processes of dielectric relaxation in a dipolar
fluid and connect them directly with thermodynamic and hydrodynamic proper-
ties, it is rather sophisticated and impractical in application. The main results are
yet to be obtained here.
A somewhat different approach to describing the generalized hydrodynamics
of dipolar systems has been recently proposed in [20]. The main idea of this ap-
proach consists in the following. When vector M(k) of the dipole density is al-
ready included into the consideration as an orientational variable, it is no longer
necessary to deal with the orientational dependence Q̂(r,Θ) ≡ {n̂, P̂ , ê, L̂} of
hydrodynamic variables. Then the basic set can be cast in the usual molecular
form Q̂(k) =
∫
V
〈Q̂(r,Θ)〉
Θ
e−ik·rdr ≡ {n(k),P (k), e(k),L(k)}, where n(k) =
∑N
i=1 e
−ik·ri, P (k) =
∑N
i=1mvie−ik·ri , e(k) =
∑N
i=1〈ei〉Θi
e−ik·ri and L(k) =
∑N
i=1
Jwie−ik·ri . The advantage of such a representation lies in the evident simplification
of the corresponding hydrodynamic equations. Moreover, owing to conservation
laws of the total number of particles, momenta and energy, the basic set Q̂(k)
can be considered as a set of indeed slow variables. For instance, time derivatives
of n(k), P (k) and e(k) vanish when k → 0. Finally, for isotropic and spatially
homogeneous systems, the dipole density M(k) does not correlate at equilibrium
with the basic hydrodynamic quantities in the static limit, i.e., 〈M(k)·Q̂(k)〉 = 0.
For this reason, the dipole-density and hydrodynamic fluctuations can be con-
sidered separately from each other as this has been assumed implicitly in the
dipole-density formalism of Madden and Kivelson [2]. The generalized collective
modes related to dipole-density fluctuations will be called dipolar modes in order
to distinguish them from the hydrodynamic modes, concerning fluctuations of hy-
drodynamic quantities. At the same time, the generalized hydrodynamic modes of
a dipolar system can be studied within the same scheme as in the case of simple
fluids [21–24]. It is also worth remarking that the correlations between the dipole-
moment density and hydrodynamic quantities are not completely independent.
They, being absent at the basic level, appear, however, at higher-orders of the
180
Dielectric relaxation in dipolar fluids
description. This leads to additional non-Markovian effects, when working within
the dipole-density formalism. Nevertheless, these effects are taken into account in
an effective way, including, besides the basic quantity M(k), its higher-order com-
ponents as well. The hydrodynamic correlations can be included in the same way,
too. Time constants appearing in the dissipative kernel are usually considered, as
adjustable parameters. It is worth stressing that the previous applications of this
theory were restricted to the long-wavelength regime only [12].
Recently, the concept of generalized collective modes, used earlier for the in-
vestigation of nonequilibrium properties of simple fluids [21–25], has been applied
to dipolar systems and actual computations have been performed in the whole
wavevector range within the three-mode description [26]. In particular, it has been
concluded that the three-variable prescription is sufficient to predict quantitatively
the frequency dependence of dielectric quantities for a Stockmayer fluid. But such
a conclusion was based on the calculations carried out with the help of a fitting
procedure, because a higher-order static correlation function was not known. For
this reason, it is not obvious that the described above pattern takes place indeed.
In the present paper, the generalized dipolar mode spectra of a Stockmayer
fluid are evaluated over a wide scale of wavelengths up to the five-variable de-
scription without involving any adjustable parameters. Within the framework of
the proposed approach, the frequency dependence of dielectric quantities is de-
termined by extended continued fractions to which the Markovian approximation
is applied. As a result, by using the dipolar modes it is shown that the three-
variable theory reproduces qualitatively the wavevector- and frequency-dependent
dielectric constant and only beginning from the five-order description one can talk
about a quantitative reproduction over the whole range of varying wavenumbers
and frequencies. Moreover, we demonstrate that within the same approximation
all times constants of memory kernels can be expressed in terms of static correla-
tion functions, so that dynamic properties of the system are obtained using static
fluctuations exclusively.
2. General theoretical framework
2.1. Basic definitions
We shall consider a dipolar fluid of N identical molecules enclosed in volume
V . Let
M(k, t) =
N
∑
i=1
µi(t)e
−ik·ri(t) (1)
be the primary dynamical variable whose correlation function is of direct in-
terest. The collective variable (1) presents, in fact, the spatial Fourier trans-
form
∫
V
M̂(r, t)e−ik·rdr of the microscopic operator of dipole-moment density
M̂(r, t) =
∑N
i=1µi(t)δ(r − ri(t)) where ri(t) and µi(t) denote the dipole moment
and position, respectively, of molecule i at time t. This dynamical variable satisfies
181
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
the equation of motion dM (k, t)/dt = LM (k, t), where
L =
N
∑
i=1
(
vi·
∂
∂ri
+wi×µi ·
∂
∂µi
)
−
N
∑
i 6=j
(
1
m
∂ϕij
∂ri
·
∂
∂vi
+
1
J
µi×
∂ϕij
∂µi
·
∂
∂wi
)
(2)
is the Liouville operator of the system, ϕij denotes an intermolecular potential,
vi and wi are the translational and rotational velocities, respectively, of molecule
i with mass m and moment of inertia J . The formal solution of the equation of
motion is M (k, t) = eLtM(k, 0) ≡ eLtM (k).
Let us consider the extended set of variables
{
M(k), LM(k), L2M(k), . . . , LS−1M (k)
}
≡
{
Lα−1M(k)
}
, α = 1, . . . , S (3)
which includes, besides the basic dynamical quantity M (k), its time derivatives
at t = 0 up to the (S − 1)-th order. It is necessary to note that due to isotropy of
the system, longitudinal and transverse fluctuations are completely independent of
one another and can be studied separately. Therefore, set (3) is formed by either
longitudinal, ML(k), or transverse, MT(k), components of M(k). For the conve-
nience of further presentation we transform the extended set to the orthogonalized
form
M (k) =
{
M1(k),M2(k),M3(k), . . . ,MS(k)
}
, (4)
where M1(k) = M(k), M2(k) =
(
1−P1
)
LM(k), . . . , MS(k) =
(
1−PS−1
)
LS−1
M(k) and Pα . . . =
∑α
β=1〈. . . ·M β(−k)〉〈Mβ(k)·Mβ(−k)〉−1Mβ(k) denotes the
Mori-like projection operator [27, 28]. The orthogonalized variables (4) constitute
the matrix of time correlation functions (TCFs)
fαβ(k, t) =
〈
Mα(k)· eLtMβ(−k)
〉
{2}Nµ2
≡ F(k, t) (5)
which is diagonal in the static limit t → 0, i.e., fαβ(k) = δαβfαα(k) where 〈 〉
denotes the equilibrium average, µ designates the permanent magnitude of the
molecule’s dipole moment and the multiplier {2} is included in the case of trans-
verse fluctuations only.
According to the construction of the orthogonalized set, the functions fαα(k)
can be found in terms of static correlation functions (SCFs) corresponding to the
initial set (3), gαβ(k) = 〈Lα−1M(k)·Lβ−1M(−k)〉/{2}Nµ2, by the iterations
ξαβ(k) = gαβ(k)−
β−1
∑
γ=1
ξαγ(k)ξβγ(k)
fγγ(k)
, fαα(k) = gαα(k)−
α−1
∑
γ=1
ξ2αγ(k)
fγγ(k)
(6)
with f11(k) = g11(k) and α = 2, 3, . . . , S; β = 1, 2, . . . , α−1, where it is understood
that the sum in the right-hand side of the first equality of (6) must be omitted at
β = 1 and ξαβ(k) = 〈Lα−1M(k)·Mβ(−k)〉/{2}Nµ2 denotes an auxiliary matrix.
182
Dielectric relaxation in dipolar fluids
The orthogonalized procedure can be simplified significantly taking into account
that the basic function g(k, t) ≡ f11(k, t) is even with respect to time. Then one
obtains that gαβ(k) (as well as ξαβ(k)) are equal to zero if α+β is an odd number,
whereas nonzero elements can be expressed via their diagonal ones as gαβ(k) =
(−1)
|α−β|
2 gγγ(k), where γ = (α+ β)/2.
The processes of dynamical polarization in the system can be described by
TCFs (5). In particular, the longitudinal εL(k, ω) and transverse εT(k, ω) com-
ponents of the wavevector- and frequency-dependent dielectric permittivity are
expressed via the first element g(k, t) of the S × S square matrix F(k, t) as [15]:
εL(k, ω)− 1
9yεL(k, ω)
= gL(k)− iωgL(k, ω) ,
εT(k, ω)− 1
9y
= gT(k)− iωgT(k, ω) , (7)
where y = 4πNµ2
/
9V kBT and kB, T are the Boltzmann constant and temperature
of the system, respectively, g(k, ω) =
∫ ∞
0
g(k, t)e−iωtdt ≡ Liω(g(k, t)) and Liω
designates the Laplace transform.
2.2. Extended continued fractions
Using the memory function formalism, the matrix equation for equilibrium
TCFs can be written as follows [21]:
∂
∂t
F(k, t)−Ω(k)F(k, t) +
∞
∫
0
Γ(k, τ)F(k, t− τ)dτ = 0 . (8)
In our case, due to the orthogonality of (4), the matrix of memory kernels Γ(k, τ)
has one nonzero element only, namely, ΓSS(k, τ) ≡ ΓS(k, τ), where
ΓS(k, τ) =
〈((1− PS)MS+1(k)· exp((1− PS)Lτ)(1− PS)MS+1(−k)〉
〈MS(k)·MS(−k)〉 (9)
and
Ω(k) =
〈M (k)·LM+(k)〉
〈M (k)·M+(k)
=
0 −1 0 0 0 ...
Ω2 0 −1 0 0 ...
0 Ω3 0 −1 0 ...
0 0 Ω4 0 −1 ...
0 0 0 Ω5 0 ...
... ... ... ... ... ...
(10)
is the frequency matrix with the elements
Ωα(k) =
〈Mα(k)·Mα(−k)〉
〈Mα−1(k)·Mα−1(−k)〉 ≡ fαα(k)
fα−1α−1(k)
. (11)
183
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
In the Laplace representation, the integro-differential equation (8) converts into
the algebraic one
[
iωI−Ω(k) + Γ(k, ω)
]
F(k, ω) = F(k) , (12)
where I is the unit matrix and F(k) = limt→0F(k, t) denotes the matrix of SCFs.
The equation (12) is now solved analytically and the result for the first element of
F(k, ω) can be cast as the S-th order continued fraction,
g(k, ω) =
g(k)
iω +
Ω2(k)
iω +
Ω3(k)
iω + ...
ΩS(k)
iω + ΓS(k, ω)
. (13)
In view of (6), more explicit expressions for the elements (11) of the frequency
matrix at S 6 5 are
Ω2(k) =
g2(k)
g(k)
,
Ω3(k) =
g(k)g4(k)− g22(k)
g(k)g2(k)
,
(14)
Ω4(k) =
g(k)
(
g2(k)g6(k)− g24(k)
)
g2(k)
(
g(k)g4(k)− g22(k)
) ,
Ω5(k) = g2(k)
(
g(k)g4(k)g8(k) + 2g2(k)g4(k)g6(k)− g22(k)g8(k)−
g(k)g26(k)− g34(k)
)/(
g(k)g4(k)− g22(k))(g2(k)g6(k)− g24(k)
)
,
where
g2s(k) =
〈LsM(k)·LsM (−k)〉
{2}Nµ2
(15)
is the static Kirkwood factor of the 2s-th order (s = 0, 1, . . . , S − 1) and g0(k) ≡
g(k).
It can be easily seen from (13) that the same function g(k, ω) is obtained
within the (S + 1)-th order continued fraction, too, if the memory functions obey
the recurrent relation
ΓS(k, ω) =
ΩS+1(k)
iω + ΓS+1(k, ω)
. (16)
The solution (13) leads to exact results at arbitrary order S, provided the cor-
responding memory functions ΓS are precisely determined. However, such a pre-
scription is rather a formal one, because the exact calculation (9) of the dissipative
kernel constitutes, in general, an unresolvable problem. We shall now consider a
question of how to perform this calculation approximately.
184
Dielectric relaxation in dipolar fluids
Let
τS(k) =
S
min
α=1
∫ ∞
0
t|fαα(k, t)|dt
∫ ∞
0
|fαα(k, t)|dt
, τS+1(k) =
∞
max
α=S+1
∫ ∞
0
t|fαα(k, t)|dt
∫ ∞
0
|fαα(k, t)|dt
(17)
be characteristic intervals of decaying in time of the autocorrelation functions
from the S-order set and all the rest functions from higher-order sets, and γS(k) =
τS+1(k)/τS(k) be their ratio. We assume in advance that there exist dynamical
processes in the system, corresponding to essentially different scales of time and,
therefore, beginning from some number S, the ratio γS must become sufficiently
small, i.e., γS(k) ≪ 1 at arbitrary wavevectors. This assumption is justified pro-
vided that the S-order set (4) forms an almost complete slow set of dynamical
variables. It is obvious that in this case the memory kernel (9), which is built on
projected higher-order variables, decays in time faster than TCFs (5), i.e.,
∞
∫
0
Γ(k, τ)F(k, t− τ)dτ ≈
∞
∫
0
Γ(k, τ)dτ F(k, t) ≡ Γ(k)F(k, t) . (18)
Thus, on characteristic time scales of varying TCFs, the dissipative kernel can be
considered as a δ-function in time space, ΓS(k, τ) ≈ ΓS(k)δ(τ), with the weight
ΓS(k) ≡ ΓS(k, ω = 0). In the frequency representation the relation (18) takes the
form:
ΓS(k, ω) ≈ ΓS(k) (19)
that represents the well-known Markovian approximation [2, 4].
The memory kernel ΓS(k) can be found in terms of elements (11) of the fre-
quency matrix and the basic wavevector-depended correlation time
τcor(k) = lim
ω→0
g(k, ω)
g(k)
=
∞
∫
0
g(k, t)
g(k)
dt , (20)
putting ω → 0 in the S-order continued fraction (13). Then we obtain at dif-
ferent orders of the description the following result: Γ1(k) = 1/τcor(k), Γ2(k) =
τcor(k)Ω2(k) and (n = 2, 3, . . .)
Γ2n−1(k) =
1
τcor
n−1
∏
l=1
Ω2(n−l)+1
Ω2(n−l)
, Γ2n(k) = τcorΩ2
n−1
∏
l=1
Ω2(n−l)+2
Ω2(n−l)+1
. (21)
It is worth mentioning that the frequency independence of the memory kernel in
the S-order description does not concern the memory functions of lower orders
which depend on frequency according to the recurrent relation (16).
And now we consider a more general formulation of the Markovian approxima-
tion in higher-order descriptions. Namely, as far as the ratio γS(k) is small enough
185
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
at a given value of S, it will remain small at higher orders, too, or will even decrease
with increasing S. If this statement indeed takes place, then not only the frequency
dependence of ΓS(k, ω) can be neglected, but also the values of ΓS(k) will begin
to be almost independent from the order S of the description. Then we can write
that ΓS+1(k) ≈ ΓS(k) at sufficiently great values of S and the basic correlation
time (20) can be excluded from our consideration. Taking into account the explicit
relations (21) and letting Γ1(k) = Γ2(k) and Γ2(k) = Γ3(k), . . . , Γ2n−1(k) = Γ2n(k)
and Γ2n(k) = Γ2n+1(k), where n = 2, 3, . . ., we obtain for the correlation time
τ
(s)
cor(k) in the s-th approximation: τ
(1)
cor (k) = 1/
√
Ω2(k), τ
(2)
cor (k) =
√
Ω3(k)/Ω2(k),
. . . , and
τ (2n−1)
cor (k) =
1√
Ω2
n−1
∏
l=1
Ω2(n−l)+1
√
Ω2(n−l)Ω2(n−l)+2
, (22)
τ (2n)cor (k) =
√
Ω3
Ω2
n−1
∏
l=1
√
Ω2(n−l)+1Ω2(n−l)+3
Ω2(n−l)+2
. (23)
Finally, substituting values (22) into expressions (21) for memory functions
within the same order of the approximation, we obtain
Γ2n−1(k) ≈
√
Ω2
n−1
∏
l=1
√
Ω2(n−l)+2
Ω2(n−l)
, Γ2n(k) ≈
√
Ω3
n−1
∏
l=1
√
Ω2(n−l)+3
Ω2(n−l)+1
. (24)
In such a way, in view of (13), (14) and (23) the frequency dependence (7) of
the dielectric constant can be reproduced using static correlation functions (15)
exclusively.
2.3. Generalized collective modes
In the preceding subsection it was shown how to obtain analytic results for
dynamical quantities in the frequency representation. However, in the Markovian
approximation, the equation (8) for TCFs can be solved analytically in time space
as well. This equation can now be written as
∂
∂t
F
M(k, t) = −T(k)FM(k, t) , (25)
where the generalized evolution operator T(k) = −Ω(k) + Γ(k) is determined by
explicit expressions (10), (11) and (21) (or (23)) for Ω(k) and Γ(k), respectively,
and F
M indicates the matrix F of TCFs calculated in the Markovian approxima-
tion.
Let Xβγ(k) be an eigenvector associated with the eigenvalue Zγ(k) of the T(k)-
matrix, i.e.,
S
∑
β=1
Tαβ(k)Xβγ(k) = Zγ(k)Xαγ(k) , (26)
186
Dielectric relaxation in dipolar fluids
where α, γ = 1, . . . , S and Tαβ designate the elements of T. Then the solution to
differential equation (24) is of the form
fM
αβ(k, t) =
S
∑
γ=1
Qαβ
γ (k)e−Zγ(k)t , (27)
i.e., each element of the F
M-matrix can be expressed as a sum of S Lorentzians
which are connected with the generalized collective modes Zγ(k). The ampli-
tudes Qαβ
γ are defined in terms of eigenvectors uniquely, using the initial condition
limt→0F
M(k, t) = F(k). The result is
Qαβ
γ (k) =
S
∑
l=1
Xαγ(k)X
−1
γl (k)flβ(k) , (28)
where the matrix X
−1 is the inverse of X ≡ {Xαβ}. The component Qαβ
γ (k) de-
scribes a partial contribution of the mode Zγ(k) to the time correlation function
fαβ(k, t). Applying the Laplace transform to equation (26) yields the following
result in the frequency space
fM
αβ(k, ω) =
S
∑
γ=1
Qαβ
γ (k)
iω + Zγ(k)
, (29)
which in the particular case of α = β = 1 can be considered as an alternative
representation of extended continued fractions (13) for g(k, ω) ≡ f11(k, ω) in the
Markovian approximation (18). The result (26) obtained in the S-mode description
allows one to evaluate the S × S matrix of longitudinal and transverse TCFs (5).
From Eqs. (8) and (24) it can be easily shown that
∫∞
0
dtFM(k, t) =
∫∞
0
dtF(k, t)
or, in other words, limω→0F
M(k, ω) = limω→0F(k, ω) and the Markovian approx-
imation leads directly to exact results in the low-frequency limit. Moreover, from
the initial condition F
M(k) = F(k) (i.e., fM
αα(k) = fαα(k), α = 0, 1, . . . , S−1) it fol-
lows that [34], if the S-mode approximation is used, time derivatives of the genuine
g(k, t) and approximated gM(k, t) functions coincide at t = 0 up to the 2(S−1)-th
order. For this reason, in a limit of S → ∞ the Markovian approximation exactly
reproduces analytical in time functions. Note, however, that due to the presence
of ”long-time tails” and other anomalies in dipolar systems, the time correlation
functions g(k, t) may be nonanalytic over a specific wavenumber range. In such a
case, they are calculated only approximately, even if the hypothetical limit S → ∞
is applied.
3. Investigation of a Stockmayer fluid
In order to verify the proposed theoretical scheme, we have considered the
Stockmayer model of polar systems using the intermolecular potential ϕij =
ϕLJ(rij) + ϕdd(rij,µi,µj), where ϕLJ(rij) = 4ǫLJ[(σLJ/rij)
12 − (σLJ/rij)
6] denotes
187
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
the Lennard-Jones (LJ) part, ϕdd=−3(µi·rij)(µj·rij)/r
5
ij + (µi·µj)/r
3
ij describes
the dipole-dipole interactions and rij = ri − rj . Our numerical calculations were
performed at the reduced density n∗ = σ3
LJ
N/V = 0.822, mean temperature T ∗=
kBT/ǫLJ = 1.147, dipole moment µ∗2 = µ2/(ǫLJσ
3
LJ
) = 3.0, the moment of inertia
J∗ = J/(mσ2
LJ
) = 0.025. The maximal order of extended continued fractions was
restricted to S = 5.
According to the basic results (13), (14), (21) and (22), the knowledge of static
correlation functions g2s(k) (15) is necessary at s = 0 ÷ 4 to investigate the fre-
quency dependence of the dielectric permittivity and evaluate the generalized dipo-
lar mode spectra up to the five-variable description. There are several possibili-
ties to define the static correlation functions. The lowest-order function (s = 0)
presents the well-known Kirkwood factor g(k) which is connected with the pair
distribution function [29] and, therefore, can be calculated using one or another
approach of the equilibrium statistical mechanics. The Kirkwood factor of the sec-
ond order (s = 1) has an analytical representation [26], gL,T2 (k) = kBT
3
( 2
J
+ k2
m
).
The higher-order functions (s = 2÷ 4) are related to the four-particle distribution
function, and it is not a simple task to predict them theoretically. Usually, they
are considered as adjustable parameters. It is obvious, however, that in such a
way these functions cannot be determined uniquely. Because of this, to avoid any
additional uncertainties in the calculation of collective modes and to observe the
convergence of continued fractions in a pure form, we shall evaluate g2s(k) using
the molecular dynamics (MD) method. Details of our computer experiment are
similar to those reported earlier [15].
We note that it is necessary to distinguish the correlation functions obtained
directly in simulations for finite samples, Gs(k), from infinite-system functions,
gs(k). As it was shown previously, additional transformations to obtain g(k) from
G(k) are necessary, namely, g(k) = (1/G(k) + D(k))−1 where D(k) takes into
account the details of simulations [15]. Moreover, the lowest-order time correlation
functions related to infinite and finite systems obey the equality
1
g(k)− iωLiω(g(k, t))
=
1
G(k)− iωLiω(G(k, t))
+D(k) . (30)
Performing the Taylor expansion of (29) over inverse frequencies at ω → ∞, it can
be easily shown that higher-order static correlation functions of the infinite system
can be defined as follows:
g2(k) = G2(k)
g4(k) = G4(k) +D(k)G2
2(k) (31)
g6(k) = G6(k) +D(k)G2(k)
(
G4(k) + g4(k)
)
g8(k) = G8(k) +D(k)
(
G4(k)g4(k) +G2(k)(G6(k) + g6(k))
)
.
The finite-system functions G(k), G2(k) and G4(k) were calculated directly in
188
Dielectric relaxation in dipolar fluids
the simulations by the definition (15) using equation (1) and the explicit expres-
sions
LM (k) =
N
∑
i=1
[
wi×µi − iµi(k·vi)
]
e−ik·ri ,
(32)
L2M (k) =
N
∑
i=1
[
ẇi×µi + ((wi·µi)wi −w2
iµi)− µi(k·vi)
2
−i(2wi×µi(k·vi) + µi(k·v̇i))
]
e−ik·ri ,
for higher-order dynamical variables, where v̇i ≡ Lvi = − 1
m
∑N
j(j 6=i) ∂ϕij/∂ri
≡ 1
m
f i and ẇi ≡ Lwi = − 1
J
∑N
j(j 6=i)µi×∂ϕij/∂µi ≡ 1
J
Ki denote the trans-
lational and rotational accelerations, respectively. In view of very complicated
structures for L3M(k) and L4M(k), the highest-order static correlation functions
G6(k) and G8(k) were evaluated numerically in terms of two- and four-fold time
derivatives of the function G4(k, t) at t = 0, i.e., G6(k) = −∂2G4(k, t)/∂t
2|t=0 and
G8(k) = ∂4G4(k, t)/∂t
4|t=0. The evaluation of G6(k) and G8(k) was carried out
with the help of a special procedure to reduce numerical errors to a minimum.
The obtained in such a way longitudinal gL2s(k) and transverse gT2s(k) com-
ponents of the infinite-system functions g2s(k) (as well as GL,T
2s (k)) are displayed
in fig. 1 at s = 0 ÷ 4 in units of τ−2s
σ , where τσ = σLJ(m/ǫLJ)
1/2. The compo-
nents gL,T2s (k), as autocorrelation static functions, are positively defined at arbitrary
wavenumbers. In the limit of great wavevectors they can be calculated analytically,
namely, lim
k→∞
gL,T2s (k) = (−1)s∂2sgG(k, t)
/
∂t2s
∣
∣
∣
t→0
, where gG(k, t) = 1
3
exp(−ak2t2)
denotes the limiting Gaussian transition of gL,T(k, t) at k → ∞ and a = kBT/2m
[26]. In particular, gL,T(k) → 1
3
, gL,T2 (k) → 2
3
ak2, gL,T4 (k) → 4a2k4, gL,T6 (k) → 40a3k6
and gL,T8 (k) → 560a4k8. It is interesting to note that higher-order functions dif-
fer from lower-order ones considerably. This indicates the existence of dynamical
processes in the system which correspond to essentially different scales of time.
The results presented in figure 1 allow one to check immediately our assump-
tion about the possibility of expressing the correlation time in terms of static
correlation functions. The corresponding calculations (22) of the correlation time
τ
(s)
cor(k) performed in different approximations (s = 1 ÷ 4), as well as the exact
values (20) obtained by the MD method, are presented in figures 2a and 2b for
the cases of longitudinal and transverse fluctuations, respectively. As we can see
from the figures, already the four-variable approximation reproduces the values
of τcor(k) not only qualitatively, but even quantitatively over the whole region of
wavenumbers.
The generalized dipolar mode spectra in two-, three-, four- and five-variable de-
scriptions are shown in figure 3. In the case of transverse fluctuations (subsets (a)
and (b) of the figure), we can clearly identify the diffusive modeD1(k) which is well
separated from all the rest of the modes over a wide wavevector range. This mode
189
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
Figure 1. Transverse (T) and longitudinal (L) components of the s-order static
Kirkwood factors (s = 0÷4) for a Stockmayer fluid at n∗ = 0.822 and T ∗ = 1.147.
The MD data for the finite system are shown as dashed curves. The infinite-system
Kirkwood factors are plotted by solid curves. Note that the transverse functions,
corresponding to finite and infinite systems, are practically indistinguishable, ex-
cepting the case s = 0.
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Dielectric relaxation in dipolar fluids
Figure 2. Transverse (a) and longitudinal (b) components of the correlation time
for the Stockmayer fluid. The results in one-, two-, three- and four-order approx-
imations are plotted by long-dashed, long-short-dashed, short-dashed and solid
curves, respectively. The exact values are presented as circles.
converges rapidly to its genuine value with increasing the order of the approxima-
tion, so that it is reproduced quantitatively at small wavenumbers already within
the two-variable description. As far as the five-variable description is used, we can
talk about the quantitative reproducing at intermediate and great wavevectors as
well. The appearance of the mode D1(k) is caused by the diffusive mechanism
of dielectric relaxation in polar systems and only this mechanism is considered
in the well-known Debye theory. Neglecting the dipole-dipole interactions, as it
was done originally by Debye, one can find that in the infinite-wavelength limit
limk→0D1(k) = 2DR, where DR is a rotational diffusion coefficient. Applying the
extended hydrodynamic approach, Bagchi and Chandra improved this result and
obtained D1(k) = (2DR+DTk
2)(1+ N
V
c(k)) where DT is the translational diffusion
coefficient and c(k) denotes a component of the spherical harmonic expansion of
the two-particle direct correlation function [4]. The latter result is valid not only
for dilute systems but also for dense gases and liquids. However, it can be used at
small wavevectors exclusively. Our scheme gives the possibility to define D1(k) in
terms of the Kirkwood factor and its higher-order components at arbitrary values
of wavenumber.
It is worth remarking that the diffusive mode, as the mode with the lowest real
part, gives the main contribution to the TCFs and dielectric quantities in almost
the whole domain of k–space, especially at small wavevectors. That is why the
single-relaxation-time approximation for dipole-moment fluctuations, ∼ e−D1(k) t,
which is used in the Debye theory, can be applied here. This approximation works
well in the overdamped limit of large times t and small frequencies ω, where the
inertial motions of the liquid molecules are not important. To describe properly
the region of intermediate values of t and ω, it is necessary to consider higher-order
modes. The next two propagating modes P2(k)± iW2(k) arise additionally begin-
191
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
Figure 3. Generalized dipolar mode spectra of the Stockmayer fluid: transverse
((a), (b)) and longitudinal ((c), (d)) modes in four- ((a), (c)) and five- ((b),
(d)) order descriptions. The pure diffusive modes, real and imaginary parts of
propagating modes are marked by the symbols D, P and W, respectively. For the
purpose of comparison the results in two- and three-mode approaches are shown
in (a), (c) and (b), (d) by the thinnest curves.
192
Dielectric relaxation in dipolar fluids
ning from the three-order approach. The three-, as well as higher-order descriptions
include explicitly the free-motion effects in terms of g2(k) and correlations due to
interactions via the torque-torque 〈
∑
i,j Ki·Kj〉 and force-force 〈
∑
i,j f i·f j〉 (at
k 6= 0) fluctuations in terms of g4(k) (see Eq. (31)). Within the four-order ap-
proximation the secondary diffusive mode D3(k) appears at small k. It splits into
two new propagating modes, P3(k) ± iW3(k), with increasing the order of the
approximation to five, whereas the previous two propagating modes are moder-
ately corrected. The four- and five-mode descriptions consider higher-order kinetic
processes which are important at very small times (very high frequencies) and
large wavevector values. The transverse propagating modes describe an oscilla-
tion behaviour of time polarization fluctuations. However, it is hard to observe
these oscillations because they damp significantly during their time periods, i.e.,
P2(k) ∼ W2(k) and P3(k) ∼ W3(k).
For the longitudinal fluctuations (subsets (c), (d) in fig. 3) the pattern is quite
different. Here, we can easily distinguish two propagating modes, P1(k)± iW1(k).
Contrary to the case of transverse fluctuations, these modes exhibit a quasiparticle
feature at not very large wavenumbers, where P1(k) ≪ W1(k). They should be as-
sociated with dipolarons [30] (analogous to the well-known plasmons in Coulomb
systems), where W1(k) and P1(k) define the frequency and damping of the dipo-
laron excitations, respectively. As one can see from the figures, the dipolaron mode
is predicted already within the two-order description which includes the correct
inertial short time behaviour of polarization fluctuations. If the dipole-dipole in-
teractions are neglected, the dipolaron frequency can be defined approximately as
limk→0W1(k) =
4πNµ2
V J
≈ 35.2 τ−1
σ that is very close to the values 27.8, 33.0, 35.5 and
31.7 τ−1
σ obtained by us at S = 2, 3, 4 and 5, respectively. The secondary oscillation
process is reproduced by the next two complex-conjugated modes P2(k)±iW2(k) at
S = 4. Finally, in the five-order description the pure diffusive mode D3(k) appears
additionally.
The inequalities P1(k)<∼ W1(k), P2(k), D3(k) can be considered as a condition of
existing the dipolaron oscillations. This condition is satisfied as far as k∗ ≡ kσLJ
<
∼ 4.
With increasing wavevector values the diffusive processes begin to dominate, espe-
cially at k∗ ∼ 7, where D3(k) is much less than all the other modes. This feature
is visible in all the orders of the approximations as well. For example, in the four-
order description the propagating modes P2(k)± iW2(k) are separated into the two
pure diffusive modes D2(k) and D3(k) within a small region near k∗ ∼ 7, where
the longitudinal component gL(k) of the Kirkwood factor has a sharp maximum
(see figure 1).
In view of the behaviour of dipolar modes, the whole region of wavevectors can
be split into several characteristic intervals. In the first one the lowest-lying dipolar
modes are well separated from the rest of the modes and this separation is observed
as long as k∗ <
∼ 2 (the so-called extended hydrodynamic regime). In this interval
the Debye-like theory can be applied to transverse dipole-moment fluctuations,
whereas the longitudinal component of the dielectric permittivity can be predicted
by two complex-conjugated dipolaron modes. In the second range of intermediate
193
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
wavenumbers, 2<
∼ k∗ <
∼ 12, all the modes are mixed in a very complicated manner
(especially in the case of longitudinal fluctuations). To describe the dynamical
behaviour of dielectric quantities in this range, involving additional higher-order
modes is necessary, excepting the subinterval 6<
∼ k∗ <
∼ 8, where the longitudinal
diffusive mode dominates over all the rest of the modes, similar to the behaviour
of transverse modes in the extended hydrodynamic regime. Finally, in the so-called
free-motion regime, k∗ >
∼ 12, all the modes tend to their own linear asymptotes to
reproduce the Gaussian time shape gG(k, t) of dipole-moment fluctuations.
Examples of the normalized time correlation functions ΦL,T(k, t) ≡ gL,T(k, t)/
gL,T(k) obtained in two-, three- and five-mode descriptions are presented and com-
pared with the MD data [15] in figure 4. As we can see from the figure, the trans-
verse component ΦT(k, t) exhibits an almost pure damped feature over a wide
range of wavenumbers. In the case of longitudinal fluctuations this is valid for
intermediate and great wavevector values only. At small wavevectors the longi-
tudinal dipole-moment fluctuations are described by strong dipolaron oscillations
with a slight damping. Such a behaviour of ΦL,T(k, t) is completely in line with
the predictions of the generalized dipolar modes approach. The transverse TCFs
are reproduced satisfactorily even within the three-mode description. The longi-
tudinal oscillations are described in this case as well, but only qualitatively. At
the same time, the approximated and genuine TCFs begin to be indistinguishable
with increasing the order of the description to five.
A similar pattern to that presented for TCFs is observed for the wavevector-
and frequency-dependent dielectric permittivity εL,T(k, ω)=ε′
L,T
(k, ω)−iε′′
L,T
(k, ω).
Our calculations, carried out in one-, two-, three- and five-order approximations
for the longitudinal εL(k, ω) and transverse εT(k, ω) components are shown in
figures 5 and 6, respectively, in comparison with the MD data of paper [15]. We
note that in the infinite-wavelength limit limk→0 εL,T(k, ω) = ε(ω). It can be easily
seen that within the Debye-like theory (S = 1) the dielectric permittivity can be
well reproduced in the hydrodynamic limit (low frequencies and wavenumbers).
With increasing wavevector and frequency values this theory fails especially in
the case of longitudinal fluctuations. In the three-mode approximation we can talk
about a qualitative description. Finally, within the five-variable approach the entire
frequency dependence of the dielectric permittivity is described quantitatively at
arbitrary wavevectors (deviations from the MD data do not exceed a few per cent).
This merely means that the five variables constitute an almost complete set of slow
quantities and the Markovian approximation begins to be almost exact. Therefore,
the extended continued fractions (13) converge rapidly with increasing the order of
the approximation and the hypothesis of an abbreviated description is in excellent
accord.
4. Conclusion
It has been established that dielectric relaxation in a dipolar fluid can be suc-
cessfully studied within the generalized mode method. The proposed approach can
194
Dielectric relaxation in dipolar fluids
Figure 4. The normalized time autocorrelation functions of the dipole-moment
fluctuations for the Stockmayer fluid at some fixed values of wavenumber, where
kmin = 2π/V 1/3 = 0.927/σ
LJ
. The MD data for longitudinal and transverse
components are shown as circles and squares. The results of two-, three- and
five-mode descriptions are plotted by the corresponding long-, short-dashed and
solid curves, respectively.
195
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
Figure 5. The frequency-dependence of the longitudinal dielectric permittivity
for the Stockmayer fluid at infinite ((a), (b)) and finite ((c)–(h)) wavelengths.
The MD data are shown by circles. The results obtained within one-, two-, three-
and five-mode descriptions are plotted by long-short-, long-, short-dashed and
solid curves, respectively.
196
Dielectric relaxation in dipolar fluids
Figure 6. The frequency-dependence of the transverse dielectric permittivity for
the Stockmayer fluid at finite wavelengths. Notations are as for fig. 5.
197
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
be considered as an extension of the three-variable theory of Madden and Kivelson
[2] to arbitrary numbers of dynamical variables. Our scheme for the computation
of dipolar modes is presented in such a form that is very convenient for actual
applications. In particular, all the necessary input quantities are, in fact, static
Kirkwood factors of different orders. The static factors can be determined by ei-
ther equilibrium theories or direct computer simulations. This has allowed us both
to avoid any fitting procedures and to evaluate the generalized dipolar-mode spec-
tra of a Stockmayer fluid over the whole scale of wavelengths within up to the
five-order description for the first time. It has been shown on the basis of direct
calculations that the five-variable theory enables one to define quantitatively the
dielectric permittivity of a dipolar fluid at arbitrary wavevector and frequency
values.
The investigation of generalized hydrodynamic modes of a Stockmayer fluid,
as well as the application of the proposed theory to more complicated interaction
site models of polar fluids will we presented in separate publications.
Acknowledgements
One of us (I.P.) would like to acknowledge financial support of the President
of Ukraine. I.M. also thanks the “Fonds für Förderung der wissenschaftlichen
Forschung” under Project P12422TPH for financial support.
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199
I.P.Omelyan, I.M.Mryglod, M.V.Tokarchuk
Діелектрична релаксація у дипольних рідинах.
Підхід узагальнених мод
І.П.Омелян, І.М.Мриглод, М.В.Токарчук
Інститут фізики конденсованих систем НАН України,
290011 м. Львів, вул. Свєнціцького 1
Отримано 28 листопада 1997 р.
Концепцiя узагальнених колективних мод, яка недавно була запpо-
понована для дослiдження пpостих piдин, застосовується заpаз до
опису пpоцесiв дiелектpичної pелаксацiї в дипольних системах. Да-
ний пiдхiд пpедставляє собою pозшиpення фоpмалiзму дипольної
густини до довiльного числа динамiчних змiнних i значень довжин
хвиль. Спектpи узагальнених дипольних мод Штокмайєpiвської piди-
ни визначено в шиpокiй областi довжин хвиль у наближенняx аж до
п’ятого поpядку. Дiелектpична спpийнятливiсть, залежна вiд хвильо-
вого вектоpа i частоти, та часова автокоpеляцiйна функцiя дипольно-
го моменту обчислюються на основi аналiтичних виpазiв, викоpисто-
вуючи дипольнi моди. Отpиманi pезультати поpiвнюються з pезуль-
татами нижчепоpядкових наближень i даними молекуляpної динамi-
ки. Показано, що п’яти-модовий опис кiлькiсно вiдтвоpює дiелек-
тpичну функцiю в усiй областi змiни хвильового вектоpа i частоти.
Ключові слова: колективні моди, діелектричні властивості,
узагальнена гідродинаміка
PACS: 67.55.Jd; 77.22.-d; 47.65
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