Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead
Contributions of different collective excitations to the density-density and heat-heat time correlation functions in pure liquids are studied within an approach of generalized collective modes. It is shown, that a kinetic relaxing mode, caused by slow density fluctuations, defines almost complete...
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Інститут фізики конденсованих систем НАН України
2001
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irk-123456789-1204752017-06-13T03:04:37Z Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead Bryk, T. Mryglod, I. Contributions of different collective excitations to the density-density and heat-heat time correlation functions in pure liquids are studied within an approach of generalized collective modes. It is shown, that a kinetic relaxing mode, caused by slow density fluctuations, defines almost completely the shape of density-density time correlation function for wavenumbers close to the main peak position of the static structure factor. Analytical threevariable model is used to explain negative amplitudes of contributions to dynamical structure factor from short-wavelength excitations. Such contributions can appear in the range of wavenumbers close to the main peak position of static structure factor. У методі узагальнених колективних мод досліджуються розділені вклади від різного типу колективних збуджень до часових кореляційних функцій “густина-густина” та “теплова густина-теплова густина” у простих рідинах. Показано, що кінетична релаксаційна мода, обумовлена повільними флюктуаціями густини, майже повністю визначає поведінку часової кореляційної функції “густина-густина” для значень хвильового вектора в області головного піку статичного структурного фактора. Аналітична тризмінна модель використана для пояснення негативних амплітуд, що описують вклади від коротко-хвильових збуджень до динамічного структурного фактора при значеннях хвильового вектора в області головного піку статичного структурного фактора. 2001 Article Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 387-405. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Lc, 61.25.Mv DOI:10.5488/CMP.4.3.387 http://dspace.nbuv.gov.ua/handle/123456789/120475 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Contributions of different collective excitations to the density-density and
heat-heat time correlation functions in pure liquids are studied within an approach
of generalized collective modes. It is shown, that a kinetic relaxing
mode, caused by slow density fluctuations, defines almost completely the
shape of density-density time correlation function for wavenumbers close
to the main peak position of the static structure factor. Analytical threevariable
model is used to explain negative amplitudes of contributions to
dynamical structure factor from short-wavelength excitations. Such contributions
can appear in the range of wavenumbers close to the main peak
position of static structure factor. |
| format |
Article |
| author |
Bryk, T. Mryglod, I. |
| spellingShingle |
Bryk, T. Mryglod, I. Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead Condensed Matter Physics |
| author_facet |
Bryk, T. Mryglod, I. |
| author_sort |
Bryk, T. |
| title |
Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead |
| title_short |
Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead |
| title_full |
Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead |
| title_fullStr |
Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead |
| title_full_unstemmed |
Generalized collective modes approach: Mode contributions to time correlation functions in liquid lead |
| title_sort |
generalized collective modes approach: mode contributions to time correlation functions in liquid lead |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120475 |
| citation_txt |
Generalized collective modes
approach: Mode contributions to time
correlation functions in liquid lead / T. Bryk, I. Mryglod // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 387-405. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT brykt generalizedcollectivemodesapproachmodecontributionstotimecorrelationfunctionsinliquidlead AT mryglodi generalizedcollectivemodesapproachmodecontributionstotimecorrelationfunctionsinliquidlead |
| first_indexed |
2025-07-08T17:56:48Z |
| last_indexed |
2025-07-08T17:56:48Z |
| _version_ |
1837102435480371200 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 3(27), pp. 387–405
Generalized collective modes
approach: Mode contributions to time
correlation functions in liquid lead
T.Bryk 1,2 , I.Mryglod 1
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Department of Chemistry, University of Houston, Houston, TX 77004, USA
Received April 11, 2001
Contributions of different collective excitations to the density-density and
heat-heat time correlation functions in pure liquids are studied within an ap-
proach of generalized collective modes. It is shown, that a kinetic relaxing
mode, caused by slow density fluctuations, defines almost completely the
shape of density-density time correlation function for wavenumbers close
to the main peak position of the static structure factor. Analytical three-
variable model is used to explain negative amplitudes of contributions to
dynamical structure factor from short-wavelength excitations. Such contri-
butions can appear in the range of wavenumbers close to the main peak
position of static structure factor.
Key words: hydrodynamics, time correlation functions, collective
excitations, liquids, mode contribution
PACS: 05.20.Jj, 61.20.Lc, 61.25.Mv
1. Introduction
Density-density time correlation functions Fnn(k, t) are the subject of primary
interest in studies of collective dynamics in liquids. Being obtained in an analytical
theory or in computer experiment the Fnn(k, t) can be used to explain the exper-
imental dynamical structure factor S(k, ω), which is, in fact, the spectral function
of Fnn(k, t) with k and ω being wavenumber and frequency, respectively. It is ob-
vious that any time correlation function, obtained in molecular dynamics (MD)
simulations, contains already in its shape all the information about various kinds of
collective excitations typical for the liquid considered. The simplest picture of differ-
ent mode contributions to the time correlation functions can be obtained within the
hydrodynamic treatment [1,2] when only the most slow processes, which correspond
c© T.Bryk, I.Mryglod 387
T.Bryk, I.Mryglod
to fluctuations of conserved dynamical variables, are taken into account. Three main
contributions to the density-density time correlation function Fnn(k, t) are observed
only in a pure liquid for that case [3,4], namely,
FH
nn(k, t)
FH
nn(k, t = 0)
=
γ − 1
γ
e−DT k2t +
1
γ
[
cos(cskt) +
(3Γ− b)k
cs
sin(cskt)
]
e−Γk2t, (1)
where cs, Γ, DT , γ, and b are the adiabatic sound velocity, sound attenuation coef-
ficient, thermal diffusivity, ratio of specific heats, and some constant, depending on
thermodynamic parameters, respectively. One can easily distinguish the contribution
to Fnn(k, t) from the purely relaxing thermodiffusive mode dT (k) with
dT (k) ≃ DTk
2, (2)
when k is small, and two oscillating contributions (symmetric and asymmetric ones)
from the propagating sound excitations z±s (k) with
z±s (k) ≃ Γk2 ± icsk (3)
in the hydrodynamic region. The asymmetric term in (1) has the leading order
∼ k and does not contribute to the static value Fnn(k, 0) because of an asymmetry
with respect to t. Thus, in the hydrodynamic region the collective dynamics of a
pure liquid is well described by the set of three hydrodynamic collective excitations,
which correspond to the most slow microscopic processes in a liquid. Note also
that, according to the Bogolubov’s hypothesis about hierarchy of relaxation times,
the expression (1) means also, that the other mode contributions to the density-
density time correlation function Fnn(k, t), caused by non-hydrodynamic collective
excitations, can be neglected in the hydrodynamic limit (such contributions should
be at least higher order of magnitude than k).
Beyond the hydrodynamic region the short-time processes become more impor-
tant in liquid dynamics. Kinetic collective modes (relaxing and propagating ones)
begin to contribute sufficiently to all time correlation functions. Therefore, beyond
the hydrodynamic region, the simple expression (1) cannot be applied to the study
of dynamical properties of liquids and, in particular, for the estimation of a col-
lective mode spectrum. As examples of kinetic excitations known in the literature,
we can mention optic-like excitations in ionic liquids, the so-called ‘fast sound’ and
‘slow sound’ modes in binary mixtures, and shear waves in transverse dynamics of
liquids. All these kinetic propagating collective modes cannot be described within
the standard hydrodynamic treatment.
One of the most efficient methods for the study of collective dynamics of liquids
in a wide range of k and ω is the approach of generalized collective modes (GCM)
[5,6]. This method is based on the concept of generalized collective excitations and
allows us to estimate the local coupling effects (at fixed k-value) between different
relaxing and propagating collective excitations, either hydrodynamic or kinetic ones.
Analytical treatment of simplified models, derived within the GCM approach, allows
one to also understand different mechanisms responsible for spectra formation in
different k-regions.
388
Generalized collective modes approach: Mode contributions
Dynamical properties of liquid Pb at temperatures 623 K and 1170 K were
studied experimentally using the thermal neutron scattering technique and reported
in [7] and [8], respectively. It was shown, that for the low-temperature state the
effect of collective excitations was visible on dynamical structure factor S(k, ω) as
Brillouin peaks up to k ≈ 1.5 Å−1, while for high-temperature lead, at 1170 K, no
side maxima were observed because of the rather wide central peak. That is why
according to [8] “a temperature study at small Q values would give an important
information on the possibility for a liquid metal to sustain these modes of motion”.
Another issue of the role played by the structure factor in dispersion law of collective
excitations in liquid Pb was raised in [8].
Recently the experimental and theoretical studies were performed on liquid Ga
[9], Li [10] and Al [11] with the purpose to find features of non-hydrodynamic be-
haviour in collective dynamics of the liquids studied. The non-hydrodynamic effects
were explained by an additional non-acoustic branch in the spectrum of collective
excitations [9] and by the existence of second relaxation time [10,11].
This study of mode contributions to the collective time correlation functions
in liquid lead is the follow-up of our recent study on spectra of collective excita-
tions in Pb [13]. The scheme of this study is the following: (i) using the numerical
results, obtained for the generalized collective modes in [13], to calculate the cor-
responding amplitudes of mode contributions to the density-density and heat-heat
time correlation functions and (ii) to compare the results of calculations with the
data expected from the hydrodynamic treatment. This allows us to conclude about
the role of non-hydrodynamic effects when wavenumber k increases. Hence, the goal
of this study is to investigate the k-dependence of the mode amplitudes, which de-
scribe the separated contributions from the generalized hydrodynamic and kinetic
collective excitations to time correlation functions in pure liquids. For this purpose
we consider a model of liquid lead at two temperatures. We focus special attention
on the role of kinetic collective modes and discuss this problem more in detail.
The paper is organized as follows. In the next section the generalized expressions
for time correlation functions, which have a wider range of application in comparison,
e.g., with (1), are derived. In section III the amplitudes of mode contributions are
studied in detail for the density-density and heat-heat time correlation functions.
We also discuss the origin of kinetic purely relaxing mode d2(k), found in our recent
GCM-studies of pure liquids [12–14]. The last section contains some concluding
remarks.
2. Time correlation functions within the GCM approach
Within the Nv-variable approximation of the GCM approach the solutions for
time correlation functions can be written (see, e.g., [15]) in an analytical form via
the eigenvalues zα(k) and eigenvectors of the generalized hydrodynamic matrix. In
this case any time correlation function, constructed on dynamical variables from the
389
T.Bryk, I.Mryglod
Nv-variable basis set, is expressed as a weighted sum of Nv mode contributions
Fij(k, t) =
Nv
∑
α=1
Gα
ij(k) exp{−zα(k)t} . (4)
Each term in the expression (4) is associated with the relevant collective excita-
tion. The k-dependent function Gα
ij(k) is the complex weight coefficient of mode
contribution to Fij(k, t), associated with the eigenvalue zα(k). In a three-variable
approximation for the hydrodynamic set AH = {n(k, t), Jl(k, t), e(k, t)} the expres-
sion (1) can easily be derived from (4), assuming k to be small. Hence, one may
conclude that in the GCM approach the well-known hydrodynamic expressions for
time correlation functions (see, e.g., [2–4]) can be properly generalized for the case
of an arbitrary number of generalized collective excitations.
Let us rewrite now the expression (4), taking into account some properties of
the eigenvalues zα(k) and amplitudes Gα
ij(k). Assuming that among Nv eigenvalues
zα(k) there are Np pairs of complex conjugated eigenvalues (propagating modes)
z±α (k),
z±α (k) = σα(k)± iωα(k), α = 1, ..., Np,
and Nr purely real ones (relaxing modes) dα(k), α = 1, ..., Nr (note that Nr =
Nv − 2Np), one can rewrite equation (4) in the form
Fij(k, t) =
Nr
∑
α=1
Aα
ij(k) e
−dα(k)t
+
Np
∑
α=1
{
Bα
ij(k) cos[ωα(k)t] + Cα
ij(k) sin[ωα(k)t]
}
e−σα(k)t. (5)
The new amplitudes Aα
ij(k), B
α
ij(k) and Cα
ij(k) in (5) are simply connected with
the functions Gα
ij(k) and depend only on wavenumber k. In further consideration we
will call the terms with the amplitudes B α
ij(k) and Cα
ij(k), caused by the propagating
modes zα(k), as symmetric and asymmetric contributions, respectively. Obviously,
Nr
∑
α=1
Aα
ij(k) +
Np
∑
α=1
Bα
ij(k) = Fij(k, t = 0). (6)
By taking Fourier transform of equation (5) one obtains the expression for a spectral
function F̃ij(k, ω) with the separated mode contributions. It is seen from (5) that
F̃ij(k, ω) will contain the contributions from Nr central Lorentzians, 2Np non-central
Lorentzians (symmetric contributions) at frequencies ±ωα, and 2Np non-Lorentzian
corrections (asymmetric contributions), respectively.
3. Results and discussion
We performed MD simulations in a standard microcanonical ensemble for liquid
Pb at two thermodynamic points: a high-temperature state at 1170 K with number
390
Generalized collective modes approach: Mode contributions
0
5
10
15
20
25
30
35
z1
z2
z3
z4
ℑ
m
z
j(k
)
(
ps
-1
)
(a) 623K
0
2
4
6
8
10
0 0.5 1 1.5 2 2.5 3
d1
d2
d3
k (A�°-1)
ℜ
e
z j
(k
)
(
ps
-1
)
0
5
10
15
20
25
30
35
40
z1
z2
z3
z4
(b) 1170K
ℑ
m
z
j(k
)
(
ps
-1
)
0
5
10
15
0 0.5 1 1.5 2 2.5 3
d1
d2
d3
k (A�°-1)
ℜ
e
z j
(k
)
(
ps
-1
)
Figure 1. Spectra of collective excitations of liquid Pb at 623 K (a) and 1170 K
(b), obtained for the nine-variable set A(9)(k, t). Complex and purely real eigen-
values are shown by symbols, connected by spline-interpolated solid and dashed
lines, respectively. The same symbols in upper and lower frames correspond to
imaginary and real parts of propagating modes, except the case of splitting of
complex-conjugated eigenvalue z1(k) into two real ones for small k-values. The
straight dash-dotted lines in upper frames show the linear hydrodynamic disper-
sion of sound excitations.
density n = 0.0289 Å−3 , and a state above melting temperature at 623 K with
number density n = 0.03094 Å−3. In molecular dynamics we studied a system of
1000 particles interacting through oscillating potential Φ ij(r) at a constant volume
V = L3. The smallest wavenumbers achieved in MD were kh
min = 0.1928 Å−1 and
kl
min = 0.1973 Å−1 for high- and low-temperature states, respectively. The time evo-
lution of hydrodynamic variables and their time derivatives was observed during the
production run over 3 ·105 steps for each temperature. The effective two-body poten-
tial was taken in an analytical form from [16]. This potential very well reproduced
the experimental static structure factor of liquid lead over a wide temperature range
[16].
3.1. Longitudinal collective excitations in liquid lead
Within the nine-variable approximation [17] of the parameter-free GCM method
the basis set of dynamical variables for the case of longitudinal dynamics in pure
liquids consists of the following operators:
A(9)(k, t) =
{
n(k, t), Jl(k, t), e(k, t), J̇l(k, t), ė(k, t), J̈l(k, t), ë(k, t),
...
Jl (k, t),
...
e (k, t)
}
,
(7)
391
T.Bryk, I.Mryglod
where n(k, t), Jl(k, t), e(k, t) are the hydrodynamic variables, namely, the densities of
particles’ number, longitudinal current and energy, respectively. The dots denote the
order of time derivative of a relevant operator. The basis set of dynamical variables is
applied to generate the eigenvalue problem from the generalized Langevin equation
in Markovian approximation [6,17].
The eigenvalues of the generalized hydrodynamic matrix T(k), generated by the
basis set (7), were calculated for liquid Pb at low- and high-temperature states. The
results are shown in figure 1a and 1b, respectively. As functions of k, these eigenval-
ues form the spectra of generalized collective excitations. It is seen in figure 1, that
for both temperatures the spectra contain in general four branches of generalized
propagating modes: three of them exist over the whole k-range considered, and one
branch, denoted as z1(k), has the propagating gap in small-k region and appears for
k larger than some temperature-dependent value kH (kH ≈ 0.2 Å−1 for Tl = 623 K
and kH ≈ 0.4 Å−1 for Th = 1170 K, respectively). Imaginary and real parts of the
complex-conjugated eigenvalues are shown in figure 1a and 1b by the same symbols
connected by spline-interpolated solid lines. Three purely real eigenvalues, marked
for convenience as dα(k) with α = 1, 2, 3, are shown in the lower frames of figure 1a
and 1b by symbols connected by spline-interpolated dotted lines. One can see, that
the purely real eigenvalues d1(k) and d3(k) exist only for small wavenumbers (inside
the propagating gap for z1(k)), while at some k-value they merged. At this k-point
the two relaxing modes disappear, and the pair of propagating excitations z±
1 (k)
emerges instead. Only one relaxing mode d2(k) exists in the whole k-region studied.
The physical meaning of this mode will be discussed more in detail below.
From the behaviour of eigenvalues at k → 0 one can establish that the pair of
propagating modes z2(k) corresponds to the generalized sound excitations with the
linear dispersion ωs(k) in hydrodynamic region. The straight dash-dotted lines in
the upper frames of figure 1a and 1b allows one to estimate the speed of longitudinal
acoustic waves in liquid Pb (see [13]).
The pair of propagating modes z±2 (k) together with the thermodiffusive mode
d1(k) form the set of generalized hydrodynamic collective excitations. All the other
eigenvalues correspond to the kinetic modes, the damping coefficients of which in
contrast to generalized hydrodynamic ones tend to some finite values when k goes
to zero, so that these excitations have the finite time of life and do not contribute
into the hydrodynamic long-time behaviour. However, as it is seen in figure 1, the
real parts of generalized hydrodynamic and kinetic modes become comparable for
intermediate and large values of k. Hence, the role of the kinetic modes increases
beyond the hydrodynamic region.
In [13] it was shown, that the propagating modes z±1 (k) are caused by heat
fluctuations and describe the low-frequency heat waves. The branch z±2 (k) is well
reproduced within the viscoelastic treatment and corresponds to generalized sound
excitations. One can also see that in the region of the main peak location of the
static structure factor k ≃ Qp the branch z±2 (k) has a minimum, which is much more
pronounced at the higher temperature. This implies that for very high temperatures
one can expect in liquid Pb a propagation gap for generalized sound excitations with
392
Generalized collective modes approach: Mode contributions
wavenumbers k ≃ Qp. The high-frequency branches z±3 (k) and z±4 (k) are mainly of
a thermal and a viscous origin, but their positions are strongly affected by lower
lying modes due to the mode-coupling effects.
3.2. Mode contributions to the density-density time correlation function
In figure 2 the leading mode contributions to the density-density time correla-
tion function, calculated for liquid lead at the higher temperature Th = 1170 K, are
shown for three values of k. These results are obtained based on our expression (4) for
time correlation functions with the separated mode contributions. The MD-derived
functions Fnn(k, t) are shown by solid lines. Dashed lines, which correspond to the
GCM functions (4), almost coincide with MD-functions, which means a very good
quality of the nine-variable approximation used in our calculations. Note that the
GCM approach does not require any adjustable or fitting parameters. Dotted lines
show the total contribution from the generalized sound excitations z2(k), i.e., the
sum of symmetric and asymmetric terms associated with the sound modes. Short-
dash-dotted lines correspond to the heat excitations: inside the propagation gap for
the heat waves, i.e., for k < kH , this line represents the contribution, caused by the
thermodiffusive relaxing mode d1(k), while for k > kH it gives the total contribution
from the heat waves z1(k) (the sum of symmetric and asymmetric terms). Purely
relaxing mode d2(k) causes the contributions shown by long-dash-dotted lines. It is
well seen in figure 2a, that in complete agreement with predictions of the hydro-
dynamic theory [1,2], the shape of density-density time correlation function for the
smallest k-value, considered in our study, is almost completely determined by the
contributions from the hydrodynamic excitations (the pair of sound excitations z2(k)
and the thermodiffusive mode d1(k)). For k-values slightly beyond the propagation
gap (k > kH) the heat waves cause a rather small negative contribution, while the
shape of function Fnn(k, t) is formed mainly by the sound excitations z2(k) and the
relaxing kinetic mode d2(k) in comparable amounts. When k increases further and
gets closer to the main peak of static structure factor k ∼ Qp, the contributions
of the low-frequency heat waves and the sound excitations become comparable and
much smaller than the term associated with the kinetic relaxing mode d 2(k), so
that the long-time tail of Fnn(k, t) in the region k ≃ Qp is completely defined by
the kinetic relaxing mode d2(k). Similar behaviour was also observed for the lower
temperature Tl = 623 K.
In order to get a more detailed picture of separated mode contributions to the
density-density time correlation function we also calculated the normalized ampli-
tudes, describing the mode contributions to the function Fnn(k, t)/Fnn(k, 0). The
results as functions of wavenumber k are plotted in figure 3 for several low-lying
collective modes (see figure 1) at two temperatures considered. Note that the weight
amplitudes of the high-frequency kinetic propagating modes z3(k) and z4(k) are
nearly two- or tree orders of magnitude smaller than the ones, caused by the sound
excitations and low-frequency heat waves. It is worth to mention that the normalized
393
T.Bryk, I.Mryglod
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0 1 2 3 4 5
F
nn
(k
,t)
t (reduced units)
(a) k=0.1928A�°-1
MD
GCM
z2
d2
d1
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.1 0.2 0.3 0.4 0.5 0.6
F
nn
(k
,t)
t (reduced units)
(b) k=0.6679A�°-1
MD
GCM
z2
z1
d2
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
F
nn
(k
,t)
t (reduced units)
(c) k=1.6809A�°-1
MD
GCM
z2
z1
d2
Figure 2. Separated mode contributions to the function Fnn(k, t) calculated for
three k-values at Th = 1170 K. The MD-derived function and the result of GCM
study are shown by solid and dashed lines, respectively. Mode contributions from
the low-frequency heat waves z1(k), the sound excitations z2(k) and the kinetic re-
laxing mode d2(k) are plotted by short dash-dotted, dotted and long dash-dotted
lines, respectively. For the smallest k-value the contribution from the thermodiffu-
sive mode d1(k) is shown by short dash-dotted line. Time scale is τh = 2.3935 ps.
394
Generalized collective modes approach: Mode contributions
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5
A
nn
(k
),
B
nn
(k
),
C
nn
(k
)
k [A-1]
s z2
d2
a z2
d3
d1
s z1
a z1
(a) 623K
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5
A
nn
(k
),
B
nn
(k
),
C
nn
(k
)
k [A-1]
(b) 1170K
Figure 3. Normalized k-dependent amplitudes of mode contributions to the
density-density time correlation function Fnn(k, t) at the temperatures Tl = 623 K
and Th = 1170 K. Lines denote the spline interpolation. Dashed and solid lines
correspond to the amplitudes of mode contributions from the relaxing and prop-
agating collective excitations, respectively.
amplitudes satisfy the following sum rule (compare with (6)):
Nr
∑
α=1
Āα
ij(k) +
Np
∑
α=1
B̄α
ij(k) = 1 . (8)
It is obvious, that the asymmetric amplitudes do not contribute to static quantities.
Note also, that due to the existence of the propagating gap for low-frequency heat
waves with a relatively small width kH , one has to distinguish two k-regions with
different numbers of relaxing and propagating modes. Three contributions from the
relaxing modes (Nr = 3) and six contributions (symmetric and asymmetric ones,
Np = 3), caused by the propagating excitations, exist for k < kH . Beyond the prop-
agation gap the situation changes and we have in (5) one contribution from the
relaxing mode and eight terms associated with the propagating modes (Nr = 1 and
Np = 4) within the nine-variable approximation (Nv = 9). For readers’ convenience
the normalized amplitudes of relaxing modes are shown in figures 3a and 3b by
symbols connected by spline-interpolated dashed lines, while symmetric/asymmetric
contributions from the propagating modes are shown by closed/open symbols con-
nected by spline-interpolated solid lines.
In general the following specific features in behaviour of the normalized mode
amplitudes can be seen in figure 3:
(i) In full agreement with the hydrodynamic theory [1] the normalized sym-
metric amplitude B̄2
nn(k), which describes the contribution from generalized sound
excitations z2(k) (solid line with closed triangles), and the amplitude Ā1
nn(k) of the
thermodiffusive mode d1(k) (dashed line with open boxes), tend to the values 1/γ
and (1 − 1/γ), respectively, when k tends to zero [13]. It is especially well seen for
higher temperature Th = 1170 K, because for the state with Tl = 623 K (just 22 K
above the melting temperature) the width of the hydrodynamic k-region is much
smaller and the smallest wavenumber kmin = 0.1973 Å−1, reached in MD in this last
case, is in fact far beyond the range of hydrodynamic behaviour. The asymmetric
395
T.Bryk, I.Mryglod
contribution C̄2
nn(k) from the acoustic excitations behaves like a linear function of
wavenumber in small-k region for Th = 1170 K, which is also in agreement with
the hydrodynamic predictions. We note that the terms associated with the kinetic
modes become negligible for small k, so that the theory is in complete consistency
with the hydrodynamic picture;
(ii) The main contribution to the shape of the density-density time correlation
functions in the region of the main peak of the static structure factor k ∼ Qp is
caused by the kinetic relaxing mode d2(k), which is of a different origin in comparison
with the thermodiffusive one d1(k). We will discuss this last statement below more
in detail;
(iii) When k increases from hydrodynamic values up to the edge of propagating
gap for the heat waves kH , one can see in figure 3, that the magnitude of mode
contributions from the heat relaxing modes d1(k) and d3(k) increase rapidly with
opposite signs and may even diverge when k → kH (in particular, such possibility is
discussed in section 3.4). On the other hand, just beyond the propagating gap, the
symmetric and asymmetric contributions from the low-frequency heat excitations
also take large enough negative and positive values, respectively. This implies, that at
the point k = kH , where two relaxing eigenvalues merge and the pair of low-frequency
heat waves appears, the corresponding amplitudes of relaxing modes diverge with
opposite signs, however, keeping their sum finite due to (8).
(iv) The behaviour of amplitudes for the heat waves in the region of wavenum-
bers k ∼ 2.7Å−1 at Th = 1170 K (figure 3b) is in some sense similar to the picture,
observed for k ∼ kH . This implies the possibility of the existence of another prop-
agating gap for the heat waves with larger k-values at higher temperatures. This
thesis is partially supported by the results for the spectra of collective excitations
(see figure 1b), where the branch z1(k) displays a rapid decay just in the region of
k ∼ 2.7 Å−1.
3.3. Origin of relaxing kinetic mode d2(k)
It was shown in previous subsection, that the kinetic relaxing mode d2(k) causes
the leading contribution to the shape of density-density time correlation function
for k ≃ Qp. This relaxing kinetic mode is well reproduced within the viscoelastic
subset of dynamical variables
A(5)(k, t) =
{
n(k, t), Jl(k, t), J̇l(k, t), J̈l(k, t),
...
Jl (k, t)
}
.
Hence, its origin is mainly viscoelastic one. This is in contrast with the hydrodynamic
behaviour (1), where the relaxing contribution to density-density time correlation
function is only due to thermodiffusive processes. To study the origin of relaxing
mode d2(k) more in detail let us consider the simplest case of dynamics when only
one dynamic variable A(1) = {n̂(k)}, namely, the particles’ density n̂(k) is taken
into account. This immediately gives the solution for the only collective mode d 0(k),
which has a very simple form:
d0(k) = τ−1
nn (k).
396
Generalized collective modes approach: Mode contributions
0
1
2
3
623K
A9
A1
0
2
4
6
8
0 0.5 1 1.5 2 2.5 3
d 2
(k
)
(p
s-1
)
k (A�°-1)
1170K
Figure 4. Behaviour of the relaxing kinetic mode d2(k) (symbols) obtained in
the nine-variable GCM approximation at the temperatures: (a) Tl = 623 K, and
(b) Th = 1170 K. The one-variable relaxing mode d0(k) is shown by spline-
interpolated dashed line with triangles.
The quantity τnn(k) is the generalized correlation time, associated with the density-
density time correlation function:
τnn(k) =
1
Fnn(k, 0)
∫
∞
0
Fnn(k, t)dt . (9)
In figure 4 we show the results for two eigenvalues d0(k) and d2(k), obtained
within the one- (line with triangles) and nine-variable (closed boxes) approximations
of the GCM approach for liquid Pb at Tl = 623 K and Th = 1170 K. In the low-
temperature state a quantitative agreement between two eigenvalues for k ∼ Qp is a
striking feature. This implies that at low temperatures the slow density fluctuations
for k-values being in the region of the main peak’s position (that corresponds in fact
to the short-range fluctuations with the average interparticle distance 〈a〉, Qp ≈
2π/〈a〉) are well separated in time (see figure 1) from the thermal processes as well
as from the fast density fluctuations, which are responsible for sound propagation
in this region, and determine almost completely the shape of the density-density
time correlation function. For the higher temperature the quantitative agreement
between the results, found for d2(k) and d0(k), is not so perfect due to the stronger
397
T.Bryk, I.Mryglod
coupling with the thermal fluctuations. However, the contribution from the mode
d2(k) to Fnn(k, 0) was still dominant when k is close to Qp (see, e.g., figure 2c for the
wavenumber k = 1.6809 Å−1). In particular, such a behaviour of the kinetic relaxing
mode d2(k) explains why the mode-coupling theory of freezing was so successful by
treating just the density fluctuations nearby the region of the main peak of static
structure factor [18,19] and completely ignoring the thermal properties.
Within the simplified one-variable theory one gets the expression for the function
Fnn(k, t) in the following single-exponential form:
F 1
nn(k, t) = G1
nn(k) exp{−d0(k)t} ≡ S(k) exp{−t/τnn(k)}. (10)
Thus, within such a treatment the function τnn(k) gives in fact the estimation for
some specific time of relaxation, which, in particular, for k = Qp has a meaning of
the lifetime for a particle in the cage of the nearest neighbors, and, therefore, the
value 1/τnn(Qp) allows us to estimate the cage diffusion coefficient.
Let us summarize the results obtained for the generalized relaxing kinetic mode
d2(k):
(i) The relaxational behaviour of Fnn(k, t) in the region of the main peak of
the static structure factor S(k) is mainly determined by a single-mode contribu-
tion associated with the relaxing kinetic mode d2(k), which gives, in fact, the main
mechanism of de Gennes slowing the density fluctuations [20];
(ii) The kinetic relaxing mode d2(k) in the region of the main peak of S(k) is
defined entirely by the density fluctuations, and the thermal fluctuations have no
effects herein. Moreover, the eigenvalue d2(k) is well reproduced for k ∼ Qp even
within the one-mode approximation;
(iii) For wavenumbers k ∼ Qp the kinetic relaxing mode d2(k) determines the
lifetime of a particle in the cage of the nearest neighbors and is directly connected
with the mechanism of cage diffusion [21]. Thus, one can call this mode a structural
relaxation mode;
(iv) In the hydrodynamic range the damping coefficient d2(k) tends to nonzero
value, and its contribution to the density-density time correlation function Fnn(k, t)
becomes negligible. However, we point out that the role of relaxing kinetic mode
d2(k) increases rapidly when k becomes larger. Our recent results, obtained for liquid
metallic Cs [14] and a semi-metallic liquid Bi [12], show obviously, that beyond the
small-k region the relaxing mode d2(k) is the lowest one and this mode makes the
leading contribution to the shape of density-density time correlation function for
intermediate and large wavenumbers in complete agreement with the results found
in this study for liquid Pb.
3.4. Analytical treatment of amplitudes: three-variable theory
Our next task is to explain the behaviour of amplitudes describing the con-
tributions from the generalized sound excitations z2(k) and relaxing kinetic mode
d2(k) to the density-density time correlation function. We have derived an analyti-
cal expression for density-density time correlation function within GCM approach,
398
Generalized collective modes approach: Mode contributions
based on three dynamical variables A(3) = {n(k, t), J(k, t), J̇(k, t)}, assuming that
this basis set would be more appropriate for the description of leading relaxing and
propagating processes in the region k ∼ Qp, than the two-variable ‘damped har-
monic oscillator’ model [22]. This means that the leading dynamical processes for
k ∼ Qp are assumed to be correctly reflected by one relaxing d(k) ≈ d2(k) and two
propagating z±(k) = σ(k) ± iω(k) ≈ z2(k) modes. The three-term analytical GCM
expression for density-density time correlation function has the following form (see
(5)):
F 3
nn(k, t)
F 3
nn(k, 0)
= Ānn(k)e
−d(k)t +
[
B̄nn(k) cos{ω(k)t}+ C̄nn(k) sin{ω(k)t}
]
e−σ(k)t (11)
with the normalized amplitudes
Ānn(k) =
σ2(k) + ω2(k)− 〈ω̄2
k〉
[d(k)− σ(k)]2 + ω2(k)
,
B̄nn(k) =
d(k)[d(k)− 2σ(k)] + 〈ω̄2
k〉
[d(k)− σ(k)]2 + ω2(k)
,
C̄nn(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, 〈ω̄2
k〉 = k2kBT/mS(k) is the second frequency moment of S(k, ω). It is seen that
depending on the ratio between d(k) and the damping coefficient σ(k) = Re z(k)
one can obtain either positive or negative amplitudes for the symmetric contribution
B̄nn(k). In particular, this explains why the normalized amplitude Ānn(k) can be
larger than unity at k ≈ Qp, which can be seen in figures 3a and 3b. It also follows
from the expression for B̄nn(k), that for strongly over-damped sound excitations,
when σ(k) ≫ d(k), and small values of second frequency moment 〈ω̄2
k〉 (heavy atoms
and low temperatures), the amplitude B̄nn(k) may be negative. Thus, in this case one
obtains the negative amplitudes of sound contribution to the density-density time
correlation function or dynamic structure factor S(k, ω) within the three-variable
model.
In the case of strong damping for the short-wavelength excitations a propaga-
tion gap can emerge at k ≈ Qp. Within the propagation gap, the model produces,
instead of the pair of propagating modes z±(k), two purely real eigenvalues d(−)(k)
and d(+)(k), so that in this case the density-density time correlation function is
represented as a sum of three relaxing contributions
F 3
nn(k, t)
F 3
nn(k, 0)
= Ānne
−d(k)t + Ā(+)
nn e−d(+)(k)t + Ā(−)
nn e−d(−)(k)t (13)
with the amplitudes given by the expressions:
Ānn(k) =
〈ω̄2
k〉+ d(−)(k)d(+)(k)
(d(−)(k)− d(k))(d(+)(k)− d(k))
,
399
T.Bryk, I.Mryglod
Ā(−)
nn (k) =
〈ω̄2
k〉+ d(k)d(+)(k)
(d(k)− d(−)(k))(d(+)(k)− d(−)(k))
,
Ā(+)
nn (k) =
〈ω̄2
k〉+ d(k)d(−)(k)
(d(k)− d(+)(k))(d(−)(k)− d(+))(k)
.
Thus, we can predict that at the inner gap boundary, when d(−)(k) and d(+)(k) tend
to the same value σg, two of the amplitudes diverge and one has
Ānn →
〈ω̄2
k〉+ σ2
g
(d− σg)2
, Ā(−)
nn → −∞, Ā(+)
nn → +∞ .
However, these two amplitudes are not independent. The zero-th order sum rule
requires that the sum of Ā−
nn(k) and Ā+
nn(k) should be finite.
At the outer gap boundary, when ω(k) → 0 and σ(k) → σg in (12), only the
asymmetric amplitude diverges, so that we find
Ānn →
〈ω̄2
k〉+ σ2
g
(d− σg)2
, B̄nn →
d2 − 2σgd− 〈ω̄2
k〉
(d− σg)2
, C̄nn → +∞ .
These results correctly describe the behaviour of the mode amplitudes observed
within a three-variable viscoelastic approximation. In general, they reflect the main
properties of amplitudes when two relaxing processes merge into a propagating one.
Therefore, one can at least qualitatively understand the divergence of amplitudes
from thermal excitations on the edge of propagating gap for low-frequency heat
waves shown in figures 3a and 3b.
3.5. Mode contributions to the heat-heat time correlation function
Within the three-variable hydrodynamic treatment, the following analytical ex-
pression for the heat-heat time correlation function Fhh(k, t) can be derived:
FH
hh(k, t)
FH
hh(k, 0)
=
1
γ
e−DT k2t +
γ − 1
γ
{
cos[cskt] +
[
Γ− a
(
1 +
1
γ
)]
k
cs
sin[cskt]
}
e−Γk2t,
(14)
where a is a thermodynamic parameter. One can see, that the hydrodynamic ampli-
tudes, describing the mode contributions from the thermodiffusive mode d1(k) and
the propagating sound excitations, are now equal to 1/γ and (1− 1/γ), respective-
ly. Thus, an opposite picture in the ratio of mode contributions from thermal and
acoustic excitations is observed for the function Fhh(k, t) in comparison with the
density-density time correlation function.
Within the nine-variable approximation of GCM approach, used in our study,
one has nine separated contributions to the heat-heat time correlation function,
associated with generalized hydrodynamic and kinetic excitations (see (5)). The
results, obtained for the leading terms contributing to Fhh(k, t) at Th = 1170 K, are
presented in figure 5 for three wavenumbers k. Again, as it has been before in the
case of the density-density time correlation function, the total contributions from the
400
Generalized collective modes approach: Mode contributions
-0.5
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5
F
hh
(k
,t)
[r
ed
uc
ed
u
ni
ts
]
t [reduced units]
k=0.1928 A-1
GCM
z2
d1
d2
d3
-0.5
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6
F
hh
(k
,t)
[r
ed
uc
ed
u
ni
ts
]
t [reduced units]
k=0.6679 A-1
GCM
z1
z2
d2
-0.5
0
0.5
1
1.5
2
2.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
F
hh
(k
,t)
[r
ed
uc
ed
u
ni
ts
]
t [reduced units]
k=1.6809 A-1
GCM
z1
z2
d2
Figure 5. Separated mode contributions to the heat-heat time correlation func-
tion Fhh(k, t) for three k-values at Th = 1170 K. The results of GCM study
are shown by dashed line. The mode contributions from the low-frequency heat
waves z1(k), the sound excitations z2(k), and the kinetic relaxing mode d2(k) are
plotted by short dash-dotted, dotted and long dash-dotted lines, respectively. For
the smallest k-value considered, the contribution from the thermodiffusive mode
d1(k) and the kinetic relaxing mode d3(k) are shown by short dash-dotted and
solid lines, respectively. Time and energy scales are τh = 2.3935 ps and εh = kBTh.
401
T.Bryk, I.Mryglod
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.5 1 1.5 2 2.5
A
hh
(k
),
B
hh
(k
),
C
hh
(k
)
k [A-1]
(a) 623K
-0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5
A
hh
(k
),
B
hh
(k
),
C
hh
(k
)
k [A-1]
(b) 1170K
Figure 6. Normalized k-dependent amplitudes of mode contributions to the heat-
heat time correlation function Fhh(k, t) at two temperatures: (a) Tl = 623 K,
and (b) Th = 1170 K. Lines denote the spline interpolation. Dashed and solid
lines correspond to the mode contributions from the relaxing and propagating
collective excitations, respectively. The same symbols are used as in figure 3a.
propagating modes z1(k) and z2(k) are shown, which are simply the sums of relevant
symmetric and asymmetric terms. One can see, that for the smallest wavenumber
k = kmin, reached in MD, (see figure 5a) the results are in good agreement with
the predictions (14) of the hydrodynamic theory. In the hydrodynamic region the
main contribution to Fhh(k, t) is caused by the relaxing thermodiffusive mode d1(k),
and the relaxing shape is modulated by the contribution from the sound excitations
(the relative magnitude of this last contribution is about 20%). Note also, that the
value of k = kmin is slightly beyond the hydrodynamic k-region. Therefore, one
can see in figure 5a a tiny short-time contribution from the kinetic relaxing mode
d3(k), which is completely formed by the heat fluctuations. The contribution from
the kinetic relaxing mode d2(k) is extremely small in this region of wavenumbers.
For smaller k we expect that the contribution from the kinetic mode d3(k) vanishes,
while the amplitudes of mode contributions from the thermodiffusive mode and
sound excitations tend to the values 1/γ and (γ − 1)/γ, respectively.
When k increases the situation changes, and the low-frequency heat waves be-
gin to play a more significant role. It is seen in figure 5b, that the contributions
to Fhh(k, t) from the structural relaxing mode d2(k) and the generalized sound ex-
citations z2(k) are of the order of magnitude 40% and 20%, respectively, and an
additional, rather strong, contribution from the low-frequency heat waves z1(k) ap-
pears. For a larger wavenumber, k = 1.6809 Å−1, (see figure 5c) only two contribu-
tions from the propagating modes (the kinetic low-frequency modes z1(k) and the
generalized sound excitations z2(k)) almost completely determine the shape of the
heat-heat time correlation function. Note that in this last case the relative magni-
tude of contribution from the heat waves is about 75%, so that these modes become
dominant for large wavenumbers.
To complete the study of mode contributions to the function Fhh(k, t), we have
performed the calculations of the k-dependent normalized amplitudes, using the
scheme, described above. The results are shown in figure 6 and can be summarized
as follows:
402
Generalized collective modes approach: Mode contributions
(i) In the hydrodynamic limit, when k → 0, all the contributions from the ki-
netic modes tend to zero and become negligible, while the amplitudes from the
hydrodynamic modes d1(k) and z2(k) go to the nonzero values 1/γ and (γ − 1)/γ,
respectively, predicted by the hydrodynamic theory;
(ii) For the time correlation function Fhh(k, t) the kinetic relaxing mode d2(k)
does not play such an important role as it is observed for the density-density time
correlation function Fnn(k, t). The only region of wavenumbers with a sufficient
contribution from d2(k) is in the vicinity of the edge of the propagation gap for
low-frequency heat waves when k ≃ kH;
(iii) In the case of liquid Pb, the low-frequency heat waves make the leading
contribution to the shape of heat-heat time correlation function Fhh(k, t) beyond
the propagation gap;
(iv) The amplitudes of the relaxing modes d1(k) and d3(k) exhibit divergent-like
behaviour (as it is observed in the case of density-density time correlation function)
in the vicinity of the edge of propagation gap for the low-frequency heat waves.
4. Conclusions
The main results of this study are as follows:
i) The kinetic relaxing mode d2(k) makes the leading contribution to the shape
of density-density time correlation function in the region of wavenumbers close to
the position k = Qp of the main peak of the static structure factor S(k). This kinetic
relaxing mode is caused by slow density fluctuations and determines the lifetime of
a particle in the cage of the nearest neighbors, which is connected with the effect of
cage diffusion. This lifetime can approximately be estimated from the behavior of
the dynamical structure factor S(k, ω) as the ratio S(k = Qp, ω = 0)/S(k = Qp);
ii) The specific feature of liquid Pb is the small value of the propagation gap,
describing by kH (kH ≈ 0.2 Å−1 for Tl = 623 K and kH ≈ 0.4 Å−1 for Th = 1170 K,
respectively). As it has been shown, the asymmetric contribution from the low-
frequency heat waves to the shape of the density-density time correlation function
(or dynamical structure factor) increases dramatically when k tends to kH from the
region of large wavenumbers. In the vicinity of the edge of propagation gap for low-
frequency heat waves, this non-hydrodynamic effect, we hope, can be the subject of
experimental studies;
iii) In a wide range of wavenumbers, the low-frequency heat waves do not con-
tribute sufficiently to the density-density time correlation function. However, in the
case of the heat-heat time correlation function Fhh(k, t) their contribution is dom-
inant for the intermediate and large wavenumbers. This is the main reason why
the oscillating behaviour can be observed in time-dependence of Fhh(k, t) for larger
wavenumbers k comparing with the case of density-density time correlation function
Fnn(k, t). This result can be especially interesting in connection with new experi-
mental possibilities of impulsive stimulated light scattering scattering technique (see,
e.g., [23]), which permits to study the density response to laser-deposited heating of
a liquid;
403
T.Bryk, I.Mryglod
iv) The three-variable analytical model is used to explain an emergence of nega-
tive symmetric amplitudes for short-wavelength sound excitations with wavenumbers
k ≈ Qp.
5. Acknowledgement
I.M. thanks for the support of the Fonds für Förderung der wissenschaftlichen
Forschung under Project P12422 TPH.
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404
Generalized collective modes approach: Mode contributions
Підхід узагальнених колективних мод: модові
вклади до часових кореляційних функцій у рідкому
свинці
Т.Брик 1,2 , І.Мриглод 1
1 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
2 Факультет хімії університету м.Хюстон, Хюстон, TX 77004, США
Отримано 11 квітня 2001 р.
У методі узагальнених колективних мод досліджуються розділені
вклади від різного типу колективних збуджень до часових кореля-
ційних функцій “густина-густина” та “теплова густина-теплова гус-
тина” у простих рідинах. Показано, що кінетична релаксаційна мо-
да, обумовлена повільними флюктуаціями густини, майже повністю
визначає поведінку часової кореляційної функції “густина-густина”
для значень хвильового вектора в області головного піку статич-
ного структурного фактора. Аналітична тризмінна модель викорис-
тана для пояснення негативних амплітуд, що описують вклади від
коротко-хвильових збуджень до динамічного структурного фактора
при значеннях хвильового вектора в області головного піку статично-
го структурного фактора.
Ключові слова: гідродинаміка, часові кореляційні функції,
колективні збудження, рідини, модові вклади
PACS: 05.20.Jj, 61.20.Lc, 61.25.Mv
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