Collective dynamics in single-particle motion for pure fluids
Single-particle time correlation function Fs(k,t) of pure fluids is studied within the generalized mode approach. Analytical expression for self-intermediate scattering function, which contains oscillating terms, is obtained for three-variable basis set of single-particle dynamic variables. The m...
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| Cite this: | Collective dynamics in single-particle motion for pure fluids / T.M. Bryk, I.M. Mryglod, A.D. Trokhymchuk // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 23-42. — Бібліогр.: 23 назв. — англ. |
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Bryk, T.M. Mryglod, I.M. Trokhymchuk, A.D. 2017-06-12T17:21:04Z 2017-06-12T17:21:04Z 2003 Collective dynamics in single-particle motion for pure fluids / T.M. Bryk, I.M. Mryglod, A.D. Trokhymchuk // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 23-42. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Lc DOI:10.5488/CMP.6.1.23 https://nasplib.isofts.kiev.ua/handle/123456789/120694 Single-particle time correlation function Fs(k,t) of pure fluids is studied within the generalized mode approach. Analytical expression for self-intermediate scattering function, which contains oscillating terms, is obtained for three-variable basis set of single-particle dynamic variables. The meaning of oscillating contributions is discussed, and it is shown, that in lowdensity fluids the oscillating contributions do not appear. Our approach, developed within the five-variable scheme, is used for the analysis of MDderived single-particle time correlation function of Lennard-Jones fluid at two densities. It is shown, that the proposed scheme allows us to reproduce perfectly the function Fs(k,t) at the whole range of wavenumbers k studied. The generalized self-diffusion coefficient as a function of k is also calculated. У методі узагальнених колективних мод досліджується одно-частинкова часова кореляційна функція Fs (k,t) простих флюїдів. Для тризмінного базового набору одно-частинкових динамічних змінних отримано аналітичний вираз для відповідної функції розсіяння, що містить осциляційні вклади. Обговорюється значення цих вкладів і показано, що вони зникають у випадку негустих флюїдів. Цей же підхід, розвинутий в п’яти-змінній схемі, використано для аналізу одно-частинкових часових кореляційних функцій, розрахованих методом молекулярної динаміки для леннард-джонсівського флюїду при двох значеннях густини. Показано, що запропонований нами метод дозволяє дуже добре відтворити функцію Fs(k,t) в усій області зміни значень хвильового вектора k , яка розглядалася. Вивчено також поведінку узагальненого коефіцієнта дифузії як функції k. I.M. thanks the Fonds zur F¨orderung der wissenschaftlichen Forschung (Austria) for financial support under Project No. P15247. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Collective dynamics in single-particle motion for pure fluids Колективна динаміка в одночастинковому русі для простих флюїдів Article published earlier |
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Collective dynamics in single-particle motion for pure fluids |
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Collective dynamics in single-particle motion for pure fluids Bryk, T.M. Mryglod, I.M. Trokhymchuk, A.D. |
| title_short |
Collective dynamics in single-particle motion for pure fluids |
| title_full |
Collective dynamics in single-particle motion for pure fluids |
| title_fullStr |
Collective dynamics in single-particle motion for pure fluids |
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Collective dynamics in single-particle motion for pure fluids |
| title_sort |
collective dynamics in single-particle motion for pure fluids |
| author |
Bryk, T.M. Mryglod, I.M. Trokhymchuk, A.D. |
| author_facet |
Bryk, T.M. Mryglod, I.M. Trokhymchuk, A.D. |
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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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Колективна динаміка в одночастинковому русі для простих флюїдів |
| description |
Single-particle time correlation function Fs(k,t) of pure fluids is studied
within the generalized mode approach. Analytical expression for self-intermediate
scattering function, which contains oscillating terms, is obtained
for three-variable basis set of single-particle dynamic variables. The meaning
of oscillating contributions is discussed, and it is shown, that in lowdensity
fluids the oscillating contributions do not appear. Our approach,
developed within the five-variable scheme, is used for the analysis of MDderived
single-particle time correlation function of Lennard-Jones fluid at
two densities. It is shown, that the proposed scheme allows us to reproduce
perfectly the function Fs(k,t) at the whole range of wavenumbers k
studied. The generalized self-diffusion coefficient as a function of k is also
calculated.
У методі узагальнених колективних мод досліджується одно-частинкова часова кореляційна функція Fs (k,t) простих флюїдів. Для тризмінного базового набору одно-частинкових динамічних змінних отримано аналітичний вираз для відповідної функції розсіяння, що містить осциляційні вклади. Обговорюється значення цих вкладів і показано, що вони зникають у випадку негустих флюїдів. Цей же підхід, розвинутий в п’яти-змінній схемі, використано для аналізу одно-частинкових часових кореляційних функцій, розрахованих методом молекулярної динаміки для леннард-джонсівського флюїду при двох значеннях густини. Показано, що запропонований нами метод дозволяє дуже добре відтворити функцію Fs(k,t) в усій області зміни значень хвильового вектора k , яка розглядалася. Вивчено також поведінку узагальненого коефіцієнта дифузії як функції k.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120694 |
| citation_txt |
Collective dynamics in single-particle motion for pure fluids / T.M. Bryk, I.M. Mryglod, A.D. Trokhymchuk // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 23-42. — Бібліогр.: 23 назв. — англ. |
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| first_indexed |
2025-11-26T01:42:47Z |
| last_indexed |
2025-11-26T01:42:47Z |
| _version_ |
1850605208859574272 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 23–42
Collective dynamics in single-particle
motion for pure fluids
T.M.Bryk 1,2 , I.M.Mryglod 2 , A.D.Trokhymchuk 2,3
1 Department of Chemistry, University of Houston, Houston, TX 77204, USA
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
3 Department of Chemistry and Biochemistry, Brigham Young University,
Provo, UT 84602, USA
Received November 4, 2002, in final form January 20, 2003
Single-particle time correlation function Fs(k, t) of pure fluids is studied
within the generalized mode approach. Analytical expression for self-inter-
mediate scattering function, which contains oscillating terms, is obtained
for three-variable basis set of single-particle dynamic variables. The mean-
ing of oscillating contributions is discussed, and it is shown, that in low-
density fluids the oscillating contributions do not appear. Our approach,
developed within the five-variable scheme, is used for the analysis of MD-
derived single-particle time correlation function of Lennard-Jones fluid at
two densities. It is shown, that the proposed scheme allows us to repro-
duce perfectly the function Fs(k, t) at the whole range of wavenumbers k
studied. The generalized self-diffusion coefficient as a function of k is also
calculated.
Key words: single-particle dynamics, Lennard-Jones fluid, velocity
autocorrelation function, diffusion
PACS: 05.20.Jj, 61.20.Lc
1. Introduction
Single-particle motion in liquids is a rather complex phenomenon, which itself
involves a variety of dynamic collective processes. This is the main reason why the
single-particle dynamics is well understood so far in hydrodynamic limit only, where
the slowest dynamic processes on large spatial scales are dominant and, therefore,
some analytical methods can be used for its explanation [1,2].
Self-intermediate scattering function Fs(k, t) and velocity autocorrelation func-
tion ψ(t) are considered now to be the most popular tools in studying single-particle
motion in liquids both in the theory and in the experiment. Fs(k, t) can be extracted
c© T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk 23
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
from neutron scattering [1–3] giving us, in particular, in the hydrodynamic limit the
information about self-diffusion coefficient D. Beyond the hydrodynamic region, the
spatial inhomogeneities in a system, like clusters of atoms or moveless obstacles,
affect the shape of function Fs(k, t), and hence an appropriate method is required
to analyse the shape of Fs(k, t) at a fixed wavenumber k, which corresponds to some
spatial scale.
Recently, in [4] the problem of single-particle dynamics was attacked by com-
parison of the numerical results obtained by molecular dynamics for liquid lithium
with the predictions of several theoretical approaches, widely used in the literature.
Namely, the expressions, derived within hydrodynamic theory, Lovesey theory [5],
mode-coupling theory as well as kinetic theory, were compared with MD-derived
single-particle time correlation functions Fs(k, t) in a wide range of wavenumbers k.
In particular, it was shown, that the mode-coupling (non-local) approach, developed
by Wahnström and Sjögren [6], gave the best agreement with the simulation data
at different wavenumbers, but even in this case a noticeable discrepancies from the
MD results were observed. The results, found within Lovesey (local) theory, exhibited
some nonrealistic oscillations at high k.
For the purpose of theoretical studies of collective dynamics of liquids in a wide
range of k and ω, an approach of generalized collective modes (GCM) has been
developed [7–9]. This method is based on the concept of generalized collective exci-
tations and allows one to take into account kinetic effects in liquid dynamics as well
as local coupling between hydrodynamic and kinetic collective excitations. The GCM
method enables us to derive a more general expression, in comparison with hydro-
dynamic one, for the collective density-density time correlation function Fnn(k, t),
which already contains the contributions from the so-called kinetic collective excita-
tions (heat waves, optic-like excitations in many-component liquids, relaxing modes
connected with structural relaxation, etc.) [10–12]. It was shown that in different
k-regions, which correspond to distinct spatial scales, different processes are respon-
sible for leading contributions to Fnn(k, t) [or, alternatively, to dynamic structure
factor S(k, ω)]. In the recent years the GCM approach has been developed in order
to investigate collective dynamics in many-component fluids [13], in magnetic liq-
uids [14,15], in semi-quantum helium [16,17], in dipolar [18,19] and site interacting
models of polar [20] fluids.
In this paper we have the aim to apply the main ideas of generalized collective
mode approach to the analysis of single-particle time correlation functions Fs(k, t),
that would allow us to study an effect of diffusive-like and oscillating modes in
single-particle motion. It will be shown, that the many-variable generalized modes
approach provides a good description of single-particle dynamics in pure liquids and
theoretical results (we will abbreviate time correlation functions derived within the
GCM approach as GM-functions) almost coincide with MD-derived time correlation
functions Fs(k, t). Another important issue reported here is the study of generalized
static diffusion coefficient D(k) = D(k, ω = 0) in a pure fluid, which follows from
the memory function formalism.
24
Collective dynamics in single-particle motion
2. Generalized mode approach
Let us start from the well-known relation between the self-intermediate scattering
function Fs(k, t) and velocity autocorrelation function ψ(t) (see, e.g., [2]):
ψ(t) =
1
3
〈vi(t)vi(0)〉 = lim
k→0
{
−
1
k2
d2
dt2
Fs(k, t)
}
, (1)
where vi(t) is the velocity of the tagged particle. This expression implies, that in
dense fluids, for which velocity autocorrelation function exhibits negative minimum
connected with a “cage effect” (back scattering of the particle due to interaction
with the cage of its nearest neighbours) [1], the function Fs(k, t) already contains
the information, hidden in its shape, about collective arrangement of the nearest
neighbors. We also note that another well-known hydrodynamic expression [1,2]:
Fs(k, t) = e−Dk2t , k → 0, (2)
where D denotes the self-diffusion coefficient, obviously does not contain any oscillat-
ing contributions. This means that a leading contribution from oscillating processes
to the function Fs(k, t) must be of the order k2 in small-k limit. It is shown below,
that in the simplest three-variable GCM scheme one can easily obtain the expres-
sions for mode contributions to Fs(k, t) which behave in the hydrodynamic limit in
full agreement with equations (1) and (2).
One of the advantages of GCM approach is the possibility to represent time-
dependent quantities such as time correlation functions via separated contributions
from the long-time (or hydrodynamic) and short-time (kinetic-like) processes. Short-
time contributions for single-particle time correlation function can be connected to
purely relaxing processes or to the processes, describing oscillatory motions of par-
ticles. By analogy with the terminology used in our previous studies of collective
dynamics in liquids [8–11], we will refer to kinetic-like excitations, being associat-
ed with some short-time processes, as kinetic single-particle modes. In contrast to
hydrodynamic long-time modes, the kinetic ones have the finite lifetime in k → 0
limit, so that they do not contribute significantly to time-dependent quantities in
the hydrodynamic region. Mathematically, the relaxing and oscillatory (kinetic and
hydrodynamic) modes can be presented by the set of corresponding purely real and
complex-conjugated dynamic eigenvalues, respectively, which are the solutions of an
eigenvalue-problem for some dynamic matrix, describing the time evolution of the
system under consideration.
The starting point for a generalized mode approach is the choice of a basis set of
dynamic variables which is further used for solving the generalized Langevin equation
(GLE): a chosen basis set of dynamic variables is used to generate a matrix form of
GLE; applying GLE scheme one can derive the matrix equation for single-particle
time correlation functions; and the highest order memory function, obtained for the
chosen basic set, is then taken in Markovian approximation. In order to illustrate
our approach let us consider the following basis set of three single-particle operators:
A
(3) = {n(k, t), ṅ(k, t), n̈(k, t)}, (3)
25
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
where
n(k, t) = eikri(t)
is the dynamic variable of single-particle density (subindex of the tagged particle is
dropped in (3) for simplicity) and overdots in (3) denote the order of time derivatives
of a relevant operator.
For the basis set A
(3) the corresponding 3× 3 generalized hydrodynamic matrix
T(k), which already contains all the dissipation processes (in terminology of the
memory function formalism the matrix T(k) is in fact a sum of the matrix of memory
functions and thus the frequency matrix [8,9]) has the following form:
T
(3)(k) = F (3)(k)[F̃ (3)(k)]−1 =
0 −1 0
0 0 −1
[ω̄4,k − ω̄2,k]τ
−1
s ω̄4,k [ω̄4,kω̄
−1
2,k − 1]τs
, (4)
where F (3)(k) = F (3)(k, t = 0) and F̃ (3)(k) = F̃ (3)(k, z = 0) denote the matrices of
static correlation functions and the Laplace transforms F̃ (3)(k, z) of time correlation
functions F (3)(k, t), constructed on the variables (3), at z = 0, respectively; ω̄4,k and
ω̄4,k are the normalized frequency moments:
ω̄4,k =
〈n̈kn̈−k〉
〈ṅkṅ−k〉
= Ω2
0 +
3kBT
m
k2, ω̄2,k =
〈ṅkṅ−k〉
〈nkn−k〉
= ω2
0 =
kBT
m
k2, (5)
and the generalized correlation time τs(k) is defined by
τs(k) =
∫
∞
0
Fs(k, t)dt. (6)
Note that Ω0 in (5) has the sense of a mean frequency of vibrations for the tagged
particle (see, e.g., [3]) and is often recalled as the Einstein frequency, due to the
analogy with the crystal model, proposed by Einstein for the lattice vibrations. This
is intuitively expected in solid-like picture for an oscillating motion of the tagged
particle as a result of the interactions with its nearest neighbours.
The difference between the standard memory function formalism [1,2,5] and the
GCM approach is twofold: (i) we consider the basis set (3) of non-orthogonal dynamic
variables as the theory input. Thus, for any k value, all the time correlation functions,
constructed on dynamic variables from the basis set, in general case of the M -
variable GCM approach are presented as a weighted sum of exponential functions
with complex arguments:
F (M)
s (k, t) =
M
∑
i=1
Gα(k)e−zα(k)t, (7)
where complex numbers zα(k) are the eigenvalues of generalized dynamic matrix
T
(M)(k), and Gα(k) denote the complex (in general case) weight coefficients, evalu-
ated via the associated eigenvectors [8,9]. For further convenience we will make a dis-
tinction between purely real eigenvalues [they will be marked as dα(k) ≡ Re (zα(k))],
26
Collective dynamics in single-particle motion
describing the relaxing processes, and complex-conjugated eigenvalues zα(k), corre-
sponding to propagating excitations. Therefore, in Fourier space, the self-dynamic
structure factor Ss(k, ω), found within the GCM approach, is just the sum of Loren-
tzian-like terms for any wavenumber; (ii) within the GCM approach we do not use
any parameterizations for correlation time τs(k). The function τs(k) is just taken
as the “reference” correlation time, which can be directly evaluated from the MD-
derived function Fs(k, t), using the definition (6); so that no fitting parameters are
needed. This gives an additional sum rule in our approach [9]. However, for the pur-
pose of an analytical treatment, the k-dependence of τs(k) can be often specified.
For instance, in the hydrodynamic limit the explicit expression for τs(k) can be de-
rived. It is also worth emphasizing that the expression for Fs(k, t), obtained within
the Lovesey model [5], is easily reproduced within the three-variable GCM scheme
with a special choice for τs(k), namely:
τs(k) =
2ω2
0 + Ω2
0
ω2
0
τL(k),
where the notations of [4] are used.
To proceed further in our analysis, we note that the normalized fourth-order
frequency moment ω̄4,k [see (5)] tends to the nonzero frequency Ω2
0 in the limit
k → 0. Under this condition it can be easily shown that among three eigenvalues of
generalized hydrodynamic matrix T(k) one obtains one purely real eigenvalue with
the asymptotic behaviour:
d(k) → Dk2, k → 0,
and two eigenvalues z±(k):
z±(k = 0) =
mDΩ2
0
2kBT
(
1 ±
√
1 −
4(kBT )2
m2D2Ω2
0
)
, (8)
which can be either purely real or complex-conjugated ones, depending on the pa-
rameters of the system. Introducing two specific times τdif and τvib:
τdif =
Dm
kBT
, τvib =
1
Ω0
,
which characterize the hydrodynamic processes, connected with the selfdiffusion, and
the processes of vibrational nature being strongly dependent on an arrangement of
the tagged particle, respectively, one can see from (8) that the oscillating modes
appear only if the inequality
τdif < 2τvib (9)
is fulfilled. In this case one has for the corresponding eigenvalues:
z±(0) = σ̃ ± iω̃, (10)
27
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
with
σ̃ =
τdif
2τ 2
vib
, ω̃ = Ω0
[
1 −
(
τdif
2τvib
)2
]1/2
being the damping coefficient and the frequency of propagating modes. It is seen
that the damping coefficient σ̃ has a nonzero value at k = 0, which is the specific
feature of kinetic modes. Note that there is also one additional typical time τcol which
describes, in fact, the mean time between collisions. Its value can be estimated as
follows:
τcol =
(
m
kBTn2/3
)1/2
,
where n = N/V is a number density, so that τcol characterizes the mean-free-motion
path of a particle in noninteracting system. Thus, one can expect that for time
intervals smaller than τcol an effective Gaussian-like behaviour of time correlation
functions can be expected.
Eigenvectors, associated with the corresponding eigenvalues, determine the am-
plitude of a particular mode contribution to the self-intermediate scattering function
F
(3)
s (k, t) (upper index means three-variable basis set). For the case of complex-
conjugated eigenvalues z±(k) = σ(k) ± ω(k), in particular, when the condition (9)
is fulfilled, one obtains for any fixed wavenumber k that:
F (3)
s (k, t) = A(k)e−d(k)t +
[
B(k) cos{ω(k)t} + C(k) sin{ω(k)t}
]
e−σ(k)t (11)
with the following expressions for amplitudes:
A(k) =
σ2(k) + ω2(k) − ω̄2,k
[d(k) − σ(k)]2 + ω2(k)
, B(k) =
d(k)[d(k) − 2σ(k)] + ω̄2,k
[d(k) − σ(k)]2 + ω2(k)
,
C(k) =
d2(k)σ(k) + d(k)[ω2(k) − σ2(k)] − ω̄2,k[d(k) − σ(k)]
{[d(k) − σ(k)]2 + ω2(k)}ω(k)
. (12)
Here we adopted the notations from [10] to mark amplitudes of relaxing contributions
as A(k), and amplitudes of symmetric and asymmetric oscillating contributions as
B(k) and C(k), respectively. In the hydrodynamic limit k → 0, the amplitude A(k)
of diffusive mode d(k) ≈ Dk2 tends to unity, while the amplitude of symmetric
oscillating contribution can be a negative function of k2 depending on the ratio
between thermal velocity and diffusion coefficient:
B(k) ≈ k2
[
kBT
m
− 2Dσ̃
]
, k → 0.
Negative amplitudes of oscillating contributions are not a specific feature of single-
particle time correlation function Fs(k, t). For the collective dynamics of liquids,
a similar result was previously found [21,22] for the contributions of over-damped
sound excitations at wavenumbers k, being close to the position of the main peak
of a static structure factor. It is worth noting that the complex-conjugated modes
28
Collective dynamics in single-particle motion
can appear for fixed k even if the condition (9) is not satisfied. This point will be
discussed more in detail hereinafter together with the numerical results obtained.
It is seen in (8) that high value of self-diffusion coefficient D can lead to the
vanishing of the oscillating contributions. If the condition (9) is not fulfilled (which
is a typical situation for gases and for low density liquids) one would obtain two real
eigenvalues instead of the pair of complex-conjugated ones. Let us consider this case
more in detail. Then, according to (7), the expression for Fs(k, t) does not contain
the oscillating contributions, and in the three-variable scheme one gets:
F (3)
s (k, t) =
3
∑
α=1
Aα(k)e−dα(k)t, (13)
where the amplitudes Aα(k) can be written as follows:
Aα(k) =
dγ(k)dν(k) − ω̄2,k
[dα(k) − dγ(k)][dα(k) − dν(k)]
(14)
with α 6= γ 6= ν. This means that only the relaxing contributions form the shape
of Fs(k, t): the purely hydrodynamic contribution corresponds to the eigenvalue
d1(k) = Dk2, describing the self-diffusion process, while the purely kinetic processes
with the lifetimes of the order τ2 ≈ d−1
2 and τ3 ≈ d−1
3 are associated with the other
two modes d2(k) and d3(k), respectively. It is easy to show, that in the limit of
small wavenumbers k, the amplitudes of short-time kinetic relaxing modes d2(k)
and d3(k) [see equation (14)] are proportional to k2, while the amplitude A1(k) of
the hydrodynamic mode d1(k) tends to 1. This is in complete agreement with the
predictions of the hydrodynamic theory.
To understand the meaning of oscillating contributions to Fs(k, t), associated
with the propagating modes (10) found in the k → 0 limit, let us apply the relation
(1) to the expression (11) with amplitudes (12). It is easy to show, that the contri-
bution from the hydrodynamic mode d2(k) to velocity autocorrelation function ψ(t)
goes to zero in the limit k → 0, and only the last two terms in (11) form, in fact,
the shape of this function when k is small, so that one obtains:
ψ(3)(t) =
{
B̃ cos ω̃t+ C̃ sin ω̃t
}
e−σ̃t, (15)
where the tilted amplitudes B̃ and C̃ are given by the expressions:
B̃ =
kBT
m
, C̃ = D
ω̃2 + σ̃2
ω̃
−
kBT
m
σ̃
ω̃
=
kBT
m
σ̃
ω̃
. (16)
One can easily check that
ψ(0) =
kBT
m
and
∫
∞
0
ψ(t)dt ≡ D.
It can be also shown that two additional explicit relations for the derivatives of
function ψ(t) at t = 0:
d
dt
ψ(t = 0) = 0,
d2
dt2
ψ(t = 0) = Ω2
0
29
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
are also fulfilled for the case considered. Note that the last four relations may be
considered as a particular example of more general expressions known as the sum
rules for the function Fs(k, t). It can be shown (see, e.g., [9]) that the expression (7),
found for Fs(k, t) in a more general case of the M -variable GCM scheme, reproduces
explicitly the first (2M − 1) frequency moments at any fixed k value. This results,
because of (1), in the correct values of frequency moments for velocity autocorre-
lation function ψ(t), found within the same approach, up to the order (2M − 4)
inclusive.
What have we learned from this simple analytical treatment performed within
the three-variable GCM scheme? First, the appearance of the oscillating terms in
Fs(k, t), which determine, in fact, the shape of velocity autocorrelation function in
the small k limit, depends on the ratio of two characteristic times τdif and τvib.
Free-particle motion is expected to be dominant only in a short time limit (t� τcol
and for large values of k. In dense liquids, the vibrational (collective) component of
motion becomes more important due to “cage effect” and this results in the oscil-
lating behaviour of velocity autocorrelation function ψ(t). In the function Fs(k, t)
the contributions from the propagating kinetic-like modes are not visible in the hy-
drodynamic limit, because the amplitude of diffusive mode d1(k) is dominant [as it
follows from (12) the ratios of corresponding amplitudes B(k)/A(k) and C(k)/A(k)
are proportional to k2]. Second, within this simple treatment, all the qualitative
features of single-particle dynamics are described correctly, so that we can expect to
achieve the quantitative agreement in higher mode approximations. Note that most
of the results presented above were previously obtained within the memory function
formalism (see, e.g., [1]). This point was mainly considered in order to testify the
GCM approach to the problem of single-particle dynamics and to establish a relation
to the theoretical models developed previously.
In our further study, to analyze MD-derived self-intermediate scattering function
Fs(k, t), the five-variable scheme of GCM approach is used. This scheme can be
considered as an extension of our previous analytical treatment with the set of five
single-particle dynamic variables:
A
(5) = {n(k, t), ṅ(k, t), n̈(k, t)
...
n (k, t),
....
n (k, t)}, (17)
chosen as the basis one. Thus, additional short-time dynamic processes can be taken
into account in an appropriate way (in comparison with the case A
(3) considered
before) which has to result in a more precise short-time analysis of the numerical
results beyond the hydrodynamic region.
3. Results and discussion
We performed MD simulations in standard microcanonical ensemble for a system
of 1000 particles in a cubic box for Lennard-Jones fluid at the temperature T ∗ =
1.706 for two densities: high n∗
h = 0.845 and low n∗
l = 0.601 ones. The parameters
of LJ potential were taken as for liquid Ar: σLJ = 3.405 Å and ε = 119.8 K.
30
Collective dynamics in single-particle motion
The smallest wavenumbers achieved in MD for low- and high-density fluids were
kl
min = 0.1557 Å−1 and kh
min = 0.1745 Å−1, respectively. The time evolution of one-
particle variables and their time derivatives (see basis set A
(5)) was observed during
production runs over 1.2 · 105 steps. Fifteen values of wavenumbers k in the range
from kmin to 4.3 Å−1 were considered in our MD study. Additional averaging over all
the possible directions of wavevectors with the same absolute value was performed.
3.1. Single-particle time correlation functions
Self-intermediate scattering functions Fs(k, t), obtained in MD, and the relevant
GM-functions F
(5)
s (k, t) found for low- and high-density fluids under consideration
are shown in figures 1(a-c) and 2(a-c), respectively. One can see, that for small
wavenumbers, the shape of Fs(k, t) can be well described by hydrodynamic single-
exponential expression (2), while for intermediate values of k, the functions Fs(k, t)
decay much faster and obviously do not resemble a single-exponential function. It
is easy to see, that the MD-derived time-correlation functions in figures 1 and 2
always tend to unity with zero value of the first order time derivative as time t goes
to zero, which was expected because of time inverse symmetry. Thus, for very small
t, the precursor of free-particle Gaussian-like behaviour is observed. Otherwise, the
hydrodynamic expression (2) reflects correctly only long-time properties and does
not reproduce the right short-time behaviour. The GM-functions (7), which gen-
eralize the hydrodynamic expression (2), do have the proper short- and long-time
behaviour. Depending on the number of dynamic variables taken into the basis set,
the GM functions, as it was mentioned before, reproduce explicitly the first frequen-
cy moments (up to the order (2M − 2) within the M -variable scheme) of dynamic
structure factor Ss(k, ω) or, equivalently, the corresponding time derivatives of the
function Fs(k, t) at t = 0. Hence, the three-term GM expressions (11) and (13)
have an identical short-time behaviour with MD-derived time correlation functions
Fs(k, t) up to the fourth order time derivatives and, in addition, the zeroth time mo-
ment due to identity of correlation times τs(k). The GM-function F
(5)
s (k, t), obtained
by using the five-variable basis set (17) and shown in figures 1 and 2 by dashed lines,
gives the correct values for the first nine frequency moments (up to the eighth order
inclusive). Moreover, the integral property (6) also has a correct value within the
scheme used.
From figures 1 and 2 one can conclude, that in a wide range of k considered, the
five-term GM-functions very well reproduce the results found for a self-intermediate
scattering function Fs(k, t) of a LJ fluid. In a small time region, the free-particle
Gaussian-like t-dependence is mimicked by the finite number of exponential terms
(five in our case) keeping the correct values for the first eight time derivatives. For
a larger t the crossover to exponential behaviour is observed.
The MD-derived velocity autocorrelation functions ψ(t) of a LJ fluid, calculated
at two densities, are shown in figure 3. For the low-density state (shown by solid
line) the function ψ(t) displays a simple decay resembling the behaviour of Fs(k, t)
at intermediate wavenumbers. In the case of the high-density state, however, the
31
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
F s
(k
,t)
k=0.3115 A�°-1
MD
GM
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
F s
(k
,t)
k=1.0789 A�°-1
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
F s
(k
,t)
t / ps
k=1.6184 A�°-1
Figure 1. Self-intermediate scattering function Fs(k, t) of a LJ fluid at low-density
state: molecular dynamics simulations (solid line) and five-variable GCM scheme
(dashed line).
32
Collective dynamics in single-particle motion
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
F s
(k
,t)
k=0.3489 A�°-1
MD
GM
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
F s
(k
,t)
k=1.5209 A�°-1
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
F s
(k
,t)
t / ps
k=2.4173 A�°-1
Figure 2. Self-intermediate scattering function Fs(k, t) of a LJ fluid at high-
density state: molecular dynamics simulations (solid line) and five-variable GCM
scheme (dashed line).
33
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
N
or
m
al
iz
ed
ψ
(t)
t / ps
low
high
Figure 3. Velocity autocorrelation function of a LJ fluid: low-density (solid line)
and high-density (dashed line).
negative minimum appears, which is usually attributed to the so-called “cage effect”
[1]. This may be interpreted as the changing of tagged particle’s velocity to the
opposite direction due to interactions with the nearest neighbours (back scattering),
so that the vibrational component in single-particle motion appears.
According to our analytical treatment of oscillating modes in single-particle mo-
tion (see (15) and (11)) one can expect the oscillating contributions not only to the
shape of velocity autocorrelation function ψ(t), but to the function Fs(k, t) as well.
3.2. Generalized modes in single-particle dynamics
Let us now analyze the spectral properties of MD-derived self-intermediate scat-
tering functions Fs(k, t) using the five-variable GM approach. In figure 4 one can see
the real and imaginary parts of dynamic eigenvalues, obtained for a single-particle
dynamics of the low-density LJ fluid. Complex eigenvalues zi(k) = σi(k) ± ωi(k)
correspond to oscillating modes in single-particle motion with the frequency ωi(k)
and damping coefficient σi(k), while the purely real eigenvalues di(k) describe the
relaxing diffusive-like process. An interesting feature of low-density state is the dif-
ferent number of complex-conjugated eigenvalues z±j (k) in the regions of small and
intermediate wavenumbers. For k > 1 Å−1, there exist two branches of oscillating
modes (shown by closed and crossed boxes). For smaller wavenumbers, the oscil-
lating mode z1(k) is not supported by the low-density LJ fluid, and two relaxing
modes d2(k) and d3(k) appear in this region instead. Such a crossover from the
kinetic relaxing modes to the kinetic propagating ones at finite wavenumbers was
predicted in section 2 using a three-variable GM approach and can be described
in this case analytically. Thus, we can conclude that due to the high value of self-
diffusion (τdif > 2τvib), the long-wave oscillatory motion of particles, associated with
the vibrational component of motion, is not supported in the low-density LJ fluid
34
Collective dynamics in single-particle motion
0
5
10
15
20
25
30
Im
z
j(k
) /
p
s-1
z2
z1
0
5
10
15
20
0 0.5 1 1.5 2 2.5 3 3.5 4
R
e
z j
(k
) /
p
s-1
d1
d2
d3
k / A�°-1
Figure 4. Frequency and damping coefficients of generalized oscillating modes in
a LJ fluid, obtained for the five-variable basis set A
(5)(k, t) at low-density state.
at small wavenumbers range. For smaller spatial scales (k > 1 Å−1), the vibrational
component becomes more important and this results in the appearance of the lower
branch of kinetic-like propagating modes.
In the case of the high-density LJ fluid (see figure 5), the lower branch of prop-
agating modes z1(k) exists in the whole k-region studied, that is the consequence
of more elastic properties of high-density fluids in comparison with the low-density
ones (τdif < 2τvib). However, taking into account the value of the damping coefficient,
found for these modes, one can conclude that such excitations with the frequencies
∼ 7 ps−1 are over-damped and may be observed only for the time scale up to 0.13 ps
(which is seen, in fact, in figure 3).
For the both cases of LJ fluids considered, the high-frequency oscillatory modes
z2(k) are found within the five-variable GCM scheme. However, the damping coef-
ficients for these modes are large enough to significantly contribute to the single-
particle dynamics in the k range considered. At both densities we found that the
long-time processes are mainly connected with the extended hydrodynamic diffusive
mode d(k) which describes the self-diffusion properties in the hydrodynamic limit
35
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
0
5
10
15
20
25
30
35
40
Im
z
j(k
) /
p
s-1
z2
z1
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
R
e
z j
(k
) /
p
s-1
d1
k / A�°-1
Figure 5. Frequency and damping coefficients of generalized oscillating modes in
a LJ fluid, obtained for the five-variable basis set A
(5)(k, t) at high-density state.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
d 1
(k
) /
p
s-1
k / A�°-1
low
high
Figure 6. Generalized diffusive mode d1(k) of a LJ fluid obtained at low- (open
circles) and high-density (closed circles). The hydrodynamic asymptotics ∼ k2
are shown by solid and dashed lines.
36
Collective dynamics in single-particle motion
k → 0. To show the right hydrodynamic behaviour of the relaxing mode d(k), this
mode is plotted in figure 6 together with the function Dk2 (shown by solid and
dashed lines for the low- and high-density fluids, respectively), where D is the self-
diffusion coefficient. The values for D, obtained from hydrodynamic behaviour of the
relaxing mode d(k), are 5.3·10−5 cm2/s and 13.6·10−5 cm2/s at high and low density,
respectively. Note that for the low density LJ fluid, the self-diffusion coefficient is
almost three times larger than for the high density state. We can also compare these
values of the coefficient D, found from the asymptotic behaviour of the relaxing mode
d(k), with the results obtained for D from the long-time properties of mean squared
displacements. In the later case we found the following values for the self-diffusion
coefficient D: 5.39 ·10−5 cm2/s at the low density and 14.06 ·10−5 cm2/s for the high
density state, which is in very good agreement with the results of GCM treatment.
In general, from figures 4–6, one can conclude that the numerical results for the
spectra of self-motion modes, obtained for the five-variable basis set A
(5), strong-
ly support the conclusions of an analytical three-variable treatment made in the
section 2.
3.3. Single-particle dynamics: Mode contributions
In the previous subsection, the spectra of generalized excitations for single-
particle dynamics of a LJ fluid were calculated at two densities and the obtained
results were analyzed in comparison with the predictions of the analytical three-
variable treatment. In order to find the answer to the question about the contri-
butions of each mode in different ranges of wavenumbers k, we also calculate the
corresponding mode amplitudes as the functions of k. This allows us to study in
more appropriate way the processes responsible for single-particle motions in differ-
ent spatial scales.
As it was shown in section 2, the amplitudes of contributions from non-hydro-
dynamic kinetic-like modes to the single-particle time correlation function Fs(k, t)
should be proportional to k2 [see (12) and (14)] in the range of small wavenumbers k.
Figures 7a and 7b plot our results obtained for the amplitudes of mode contributions
from the diffusive mode d(k) (figure 7a) and from the oscillating ones (figure 7b).
These amplitudes were calculated within the five-variable GCM scheme for the basis
set A
(5) using the MD-derived functions needed. It is seen in figure 7 that, as it
was expected, the main contribution to the shape of self-intermediate scattering
function goes from the extended hydrodynamic mode d(k). The contribution of the
high-frequency pair of propagating modes z2(k) is very small even in comparison
with the amplitude of z1(k). One can also see that, in complete agreement with our
predictions, the amplitudes B(k) have the asymptote ∼ k2 in k → 0 limit, while
the amplitude of diffusive mode d(k) tends to unity in that limit. Note that in the
range k < 2.3 Å−1, the diffusive mode amplitude d(k) is larger than unity, while the
symmetric contribution from the oscillating mode z1(k) is negative. The possibility
for such a behaviour of an amplitude B(k), associated with the kinetic-like modes,
was predicted within the analytical treatment in section 2.
37
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
A
(k
)
d
-0.02
-0.01
0
0.01
0.02
0 0.5 1 1.5 2 2.5 3
B
i(k
)
k / A�°-1
z1
z2
Figure 7. Amplitudes of mode contributions to Fs(k, t) for the high-density LJ
fluid: (a) contribution A(k) of the diffusive mode and (b) symmetric oscillat-
ing contributions B1(k), B2(k) of propagating modes z1(k), z2(k). The long-
wavelength asymptotes are shown by dashed lines.
3.4. Generalized self-diffusion coefficient
The generalized (k, ω)-dependent transport coefficients often allow one to con-
sider the specific features of transport phenomena on different spatial and time
scales. Let us introduce the generalized self-diffusion coefficient Ds(k, ω) in com-
plete analogy as it was done previously with the generalized shear viscosity η(k, ω)
[23]. The following general relation between the relevance memory function and the
generalized diffusion coefficient D̃s(k, z),
D̃s(k, z = iω) = D
′
s(k, ω) − iD
′′
s (k, ω) =
1
k2
ϕ̃s(k, z = iω), (18)
can be used, where ϕ̃s(k, z) is the Laplace-transform of the lowest-order memory
function for Fs(k, t). The memory function can be evaluated from MD-derived self-
intermediate scattering function Fs(k, t), namely:
ϕ̃s(k, z = iω) =
1
∫
∞
0
e−iωtFs(k, t)dt
− iω. (19)
38
Collective dynamics in single-particle motion
2
4
6
8
10
12
14
16
0 0.5 1 1.5 2 2.5 3 3.5 4
D
(k
) /
1
0-5
c
m
2 s
-1
k / A�°-1
low
high
Figure 8. Generalized self-diffusion coefficient D(k) of a LJ fluid, calculated at
low- (open circles) and high-density (closed circles). Values at k = 0 correspond
to estimates found from long-time asymptotics of mean-squared displacements.
In this study we report only the generalized static self-diffusion coefficient, which is
obtained from (18) in static limit ω = 0, so that, using (19) and (6), one gets:
Ds(k) = D
′
s(k, ω = 0) =
1
k2τs(k)
(20)
with τs(k) being the generalized correlation time. Figure 8 plots the results, ob-
tained for the generalized self-diffusion coefficient Ds(k) of a pure LJ fluid at two
considered densities. As it was mentioned before, the values at k = 0 were also
estimated from the long-time behaviour of the mean squared displacements. The
evaluated values are in a very good agreement with the small-k behaviour of the
generalized coefficient Ds(k) obtained from equation (20). Note that the function
Ds(k) has a visible tendency to go to a finite constant when k → 0. This means
that the generalized correlation time τs(k), calculated in our MD experiment, has
the right asymptotics (proportional to 1/k2), predicted by the hydrodynamic the-
ory. In the large wavenumbers limit (or, in other words, in the free-particle limit,
where Gaussian-like approximation for Fs(k, t) can be used) it is expected [3] that
the generalized diffusion coefficient Ds(k) behaves like ∼ 1/k. Our results for Ds(k)
shown in figure 8 support this prediction for both states considered. This allows us
to conclude that in a large k domain the collisional processes with short-time scale
(less that τcol) are strongly dominating.
4. Conclusions
The main results of this study are as follows:
(I) The GCM approach is developed for the study of single-particle dynamics
in a pure liquid. We start from the simple three-variable scheme and within
39
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
the analytical treatment testify our method by comparing with the results
known previously from the memory function formalism. It is shown that two
kinds of processes determine mainly one-particle dynamic properties in a dense
fluid. One of them is connected with long-time diffusive motion, and another
reflects the elastic properties of a fluid, caused by interactions with the nearest
neighbours of the tagged particle – vibrational component of motion or the
so-called “cage effect”, which reflects mainly collective properties of a fluid.
(II) Within the GCM approach, each dynamical process can be associated with
the relevant excitation, and the spectrum of these excitations can be obtained
by solving the eigenvalue-problem for the dynamic matrix, defined for some
basis set of dynamic variables. In the three-variable scheme it is shown that in
addition to the diffusive-like mode there are the so-called kinetic-like modes,
which may describe the oscillating (collective) motions in single-particle dy-
namics. We found that the amplitudes of kinetic modes are proportional to
k2, so that it is difficult to observe their contribution to the self-intermediate
scattering function Fs(k, t), but, otherwise, this contribution is dominant in
the velocity autocorrelation function ψ(t).
(III) Within the three-variable GCM scheme, the crossover from the relaxing-like
behaviour of ψ(t), usually observed in gases and low-density fluids, to the oscil-
lating solid-like behaviour of ψ(t), typical of high-density liquids, is described.
We also derived the condition which allows us to distinguish these two cases. It
is shown that the existence of an oscillating component in ψ(t) is determined
by the value of self-diffusion coefficient Ds and the elastic properties of a flu-
id, connected mainly with the value of the so-called Einstein frequency. Our
theoretical findings are supported by numerical calculations performed within
the five-variable GCM scheme for a LJ liquid.
(IV) In order to verify our conclusions, made within the analytical treatment, we
compare the results, obtained in the five-variable GCM scheme, with the MD
simulations, performed for a LJ fluid at two densities. A very good agreement
between the both groups of results is found in a wide range of wavenumbers
k considered. It is shown, that the numerical analysis strongly supports our
analytical results. In particular, the amplitudes, describing the contributions
of kinetic modes to the function Fs(k, t), display the k2-dependence.
(V) The generalized k-dependent self-diffusion coefficient Ds(k) of a LJ fluid is
studied at both densities considered. We found that the values of Ds(k) at
k → 0 are in good agreement with the data obtained for D from long-
wavelength asymptotics of generalized hydrodynamic relaxing modes d(k). For
large wavenumbers k, the free-particle-like behaviour is observed.
An interesting issue, which has to be considered more in detail, is the relation
between oscillating single-particle modes and propagating collective excitations, ob-
served in the collective dynamics of fluids. This problem will be the subject of our
next study.
40
Collective dynamics in single-particle motion
Acknowledgements
I.M. thanks the Fonds zur Förderung der wissenschaftlichen Forschung (Austria)
for financial support under Project No. P15247.
References
1. Boon J.-P., Yip S. Molecular Hydrodynamics. New-York, McGraw-Hill, 1980.
2. Hansen J.-P., McDonald I.R. Theory of Simple Liquids. London, Academic, 1986.
3. Balucani U., Zoppi M. Dynamics of the Liquid State. Oxford, Clarendon, 1994.
4. Canales M., Padro J.A. // Phys. Rev. E, 2001, vol. 63, p. 011207.
5. Lovesey S.W. // J. Phys. C, 1973, vol. 6, p. 1856.
6. Wahnström G., Sjögren L. // J. Phys. C, 1982, vol. 15, p. 401.
7. de Schepper I.M., Cohen E.G.D., Bruin C., Rijs J.C., Montfrooij W., Graaf L.A. //
Phys. Rev. A, 1988, vol. 38, p. 271.
8. Mryglod I.M., Omelyan I.P., Tokarchuk M.V. // Mol. Phys., 1995, vol. 84, p. 235.
9. Mryglod I.M. // Condens. Matter Phys., 1998, vol. 1, p. 753.
10. Bryk T., Mryglod I. // J. Phys.: Cond. Matt., 2000, vol. 12, p. 6063.
11. Bryk T., Mryglod I. // Phys. Rev. E, 2001, vol. 63, p. 051202.
12. Bryk T., Mryglod I. // J. Phys.: Cond. Matt., 2001, vol. 13, p. 1343.
13. Mryglod I. // Condens. Matter Phys., 1997, vol. 10, p. 115.
14. Mryglod I., Folk R. // Physica A, 1996, vol. 234, p. 129.
15. Mryglod I., Folk R., Dubyk S., Rudavskii Yu. // Physica A, 2000, vol. 277, p. 389.
16. Ignatyuk V.V., Mryglod I.M., Tokarchuk M.V. // Low Temp. Phys., 1999, vol. 25,
p. 857.
17. Ignatyuk V.V., Mryglod I.M., Tokarchuk M.V. // J. Mol. Liquids., 2001, vol. 93, p. 65.
18. Omelyan I.P. // Physica A, 1997, vol. 247, p. 121.
19. Omelyan I.P., Mryglod I.M., Tokarchuk M.V. // Phys. Rev. E, 1998, vol. 57, p. 6667.
20. Omelyan I.P., Tokarchuk M.V. // J. Phys.: Cond. Matt., 2000, vol. 12, p. L505.
21. de Schepper I.M., Verkerk P., van Well A.A., de Graaf L.A. // Phys. Rev. Lett., 1983,
vol. 50, p. 974.
22. Bryk T., Mryglod I. // Phys. Rev. E, 2001, vol. 64, p. 032202.
23. Bryk T., Mryglod I. // J. Phys. Stud., 1998, vol. 2, p. 322.
41
T.M.Bryk, I.M.Mryglod, A.D.Trokhymchuk
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