Heat capacity of liquids: A hydrodynamic approach
We study autocorrelation functions of energy, heat and entropy densities obtained by molecular dynamics simulations of supercritical Ar and compare them with the predictions of the hydrodynamic theory. It is shown that the predicted by the hydrodynamic theory single-exponential shape of the entropy...
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irk-123456789-1535102019-06-15T01:27:37Z Heat capacity of liquids: A hydrodynamic approach Bryk, T. Scopigno, T. Ruocco, G. We study autocorrelation functions of energy, heat and entropy densities obtained by molecular dynamics simulations of supercritical Ar and compare them with the predictions of the hydrodynamic theory. It is shown that the predicted by the hydrodynamic theory single-exponential shape of the entropy density autocorrelation functions is perfectly reproduced for small wave numbers by the molecular dynamics simulations and permits the calculation of the wavenumber-dependent specific heat at constant pressure. The estimated wavenumber-dependent specific heats at constant volume and pressure, Cv(k) and Cp(k), are shown to be in the long-wavelength limit in good agreement with the macroscopic experimental values of Cv and Cp for the studied thermodynamic points of supercritical Ar. Ми дослiджуємо автокореляцiйнi функцiї густин енергiї, тепла та ентропiї, обчислених при моделюваннi методом молекулярної динамiки для надкритичного Ar та порiвнюємо їх iз передбаченнями гiдродинамiчної теорiї. Показано, що передбачена гiдродинамiчною теорiєю одноекспонентна форма автокореляцiйної функцiї густини ентропiї чудово вiдтворюється при малих хвильових числах молекулярною динамiкою та дозволяє розрахунок залежної вiд хвильового числа питомої теплоємностi при сталому тиску. Отриманi питомi теплоємностi, залежнi вiд хвильового числа, при сталому об’ємi та тиску, Cv (k) and Cp (k), у довгохвильовiй границi добре узгоджуються з макроскопiчними експериментальними значеннями Cv and Cp для дослiджених термодинамiчних точок надкритичного Ar. 2015 Article Heat capacity of liquids: A hydrodynamic approach / T. Bryk, T. Scopigno, G. Ruocco // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13606: 1–8. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 65.20.De, 61.20.Lc, 66.10.cd DOI:10.5488/CMP.18.13606 arXiv:1504.01224 http://dspace.nbuv.gov.ua/handle/123456789/153510 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We study autocorrelation functions of energy, heat and entropy densities obtained by molecular dynamics simulations of supercritical Ar and compare them with the predictions of the hydrodynamic theory. It is shown that the predicted by the hydrodynamic theory single-exponential shape of the entropy density autocorrelation functions is perfectly reproduced for small wave numbers by the molecular dynamics simulations and permits the calculation of the wavenumber-dependent specific heat at constant pressure. The estimated wavenumber-dependent specific heats at constant volume and pressure, Cv(k) and Cp(k), are shown to be in the long-wavelength limit in good agreement with the macroscopic experimental values of Cv and Cp for the studied thermodynamic points of supercritical Ar. |
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Bryk, T. Scopigno, T. Ruocco, G. |
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Bryk, T. Scopigno, T. Ruocco, G. Heat capacity of liquids: A hydrodynamic approach Condensed Matter Physics |
| author_facet |
Bryk, T. Scopigno, T. Ruocco, G. |
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Bryk, T. |
| title |
Heat capacity of liquids: A hydrodynamic approach |
| title_short |
Heat capacity of liquids: A hydrodynamic approach |
| title_full |
Heat capacity of liquids: A hydrodynamic approach |
| title_fullStr |
Heat capacity of liquids: A hydrodynamic approach |
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Heat capacity of liquids: A hydrodynamic approach |
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heat capacity of liquids: a hydrodynamic approach |
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Інститут фізики конденсованих систем НАН України |
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2015 |
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| citation_txt |
Heat capacity of liquids: A hydrodynamic approach / T. Bryk, T. Scopigno, G. Ruocco // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13606: 1–8. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT brykt heatcapacityofliquidsahydrodynamicapproach AT scopignot heatcapacityofliquidsahydrodynamicapproach AT ruoccog heatcapacityofliquidsahydrodynamicapproach |
| first_indexed |
2025-07-14T04:38:39Z |
| last_indexed |
2025-07-14T04:38:39Z |
| _version_ |
1837595801263538176 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 1, 13606: 1–8
DOI: 10.5488/CMP.18.13606
http://www.icmp.lviv.ua/journal
Heat capacity of liquids: A hydrodynamic approach
T. Bryk1,2, T. Scopigno3 , G. Ruocco3,4
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University,
79013 Lviv, Ukraine
3 Dipartimento di Fisica, Universita di Roma La Sapienza, I–00185, Roma, Italy
4 Center for Life Nano Science @Sapienza, Istituto Italiano di Tecnologia,
295 Viale Regina Elena, I–00161, Roma, Italy
Received March 3, 2015
We study autocorrelation functions of energy, heat and entropy densities obtained by molecular dynamics sim-
ulations of supercritical Ar and compare them with the predictions of the hydrodynamic theory. It is shown
that the predicted by the hydrodynamic theory single-exponential shape of the entropy density autocorrelation
functions is perfectly reproduced for small wave numbers by the molecular dynamics simulations and permits
the calculation of the wavenumber-dependent specific heat at constant pressure. The estimated wavenumber-
dependent specific heats at constant volume and pressure, Cv(k) and Cp(k), are shown to be in the long-
wavelength limit in good agreement with the macroscopic experimental values of Cv and Cp for the studied
thermodynamic points of supercritical Ar.
Key words: liquids, thermodynamics, specific heat, hydrodynamic theory
PACS: 65.20.De, 61.20.Lc, 66.10.cd
1. Introduction
Thermodynamics of the liquid state of matter was successfully explained on the basis of thermody-
namic perturbation theory (TDPT) more than 40 years ago [1–4]. Together with the integral equation
theory of static structure of liquids and computer simulations, it is the basis of current exploration of
physics and chemistry of real liquid systems. Modern versions of TDPT [5, 6] were recently developed on
the basis of treatment of the interaction interatomic potential as a sum of an improved reference system
(hard-sphere- or soft-sphere-like repulsion plus short-range Yukawa-like attraction) and a perturbation
due to the remaining tail of the interparticle interaction.
It is also possible to represent the thermodynamics of the liquid state via time-dependent processes
because dynamic processes in condensed matter systems are tightly connected with their thermodynam-
ics. However, in contrast to solids, where thermodynamics can be easily defined by contributions from
static lattice and small vibrations around the lattice sites in terms of phonon excitations [7], in fluids
such an approach is not applicable due to the dynamic disorder and the absence of stable local energy
minima for atomic particles. Furthermore, in liquid state there exist various relaxation processes which
are absent in the solid state. As an example one can consider a relaxation connected with diffusivity of
local temperature. In liquid system without any external temperature gradient, this relaxation process
is caused by local temperature gradients which emerge due to adiabatic propagation of the most long-
wavelength acoustic excitations. The acoustic modes, which propagate with small wavenumbers due to
local conservation laws, in fact create adiabatically compressed and decompressed local regions in liquid
with different local temperatures. Such a complex origin of collective dynamics in fluids which is com-
pletely different from the elastic sound propagation in solids is tightly connected with the phenomenon
© T. Bryk, T. Scopigno, G. Ruocco, 2015 13606-1
http://dx.doi.org/10.5488/CMP.18.13606
http://www.icmp.lviv.ua/journal
T. Bryk, T. Scopigno, G. Ruocco
of viscoelasticity of liquids [8]. Furthermore, the acoustic excitations in liquid state are only of longitu-
dinal polarization (transverse long-wavelength acoustic modes are not supported in liquids, there can
exist only short-wavelength shear excitations with a propagation gap in the long-wavelength region) and
have non-zero damping. The dispersion law of longitudinal and transverse collective excitations in liq-
uids cannot be considered as the linear in wave numbers [9, 10] as this is taken in the Debye model of
the heat capacity of solids. All these facts must be reflected in the construction of liquid thermodynamics
represented via the dynamic processes.
Heat capacity being one of the important quantities in thermodynamics of the liquid state of matter
can be easily calculated from the temperature fluctuations in the liquid system [11]. This is by date the
most simple way of estimation of the specific heat at constant volume Cv for realistic liquids from com-
puter simulations. Schofield [12, 13] has shown how the generalized wavenumber-dependent thermody-
namic quantities can be estimated from static correlators of different statistically independent dynamic
variables [14]. The generalized wavenumber-dependent thermodynamic quantities were studied later on
for pure [15–17] and binary liquids [18] in classical and ab initio [19] molecular dynamics simulations.
Perhaps the first application of the hydrodynamic theory to collective dynamics of liquids was re-
ported in the paper by Landau and Placzek [20], while systematic theoretical studies of liquid dynamics
with hydrodynamic equations have started since the seminal studies by Mountain [21]. In 1971 Cohen
et al. derived analytical expressions for hydrodynamic time correlation functions for pure and binary
liquids [22], which contained the so-called “non-Lorentzian corrections” and satisfied all the sum rules
existing within the hydrodynamic treatment [23]. Molecular dynamics (MD) simulations being a perfect
tool for exploration of time-dependent correlations in realistic liquids and solids enable one to check the
predictions of the hydrodynamic theory.
In this study we report the behaviour of autocorrelation functions of energy, heat and entropy densi-
ties obtained by molecular dynamics simulations of supercritical Ar at two densities. By date there were
no MD studies of the entropy density autocorrelation functions of pure liquids, for which the hydrody-
namics predicts purely single-exponential decay with time in the long-wavelength region. Besides, the
zero-time value of the entropy density and heat density autocorrelation functions should be connected
with the wavenumber-dependent specific heats Cp and Cv, respectively. We will perform a check of the
long-wavelength limits of the Cp(k) and Cv(k) with the macroscopic experimental values for these ther-
modynamic quantities [24]. The remaining part of the paper is organised as follows. In the next section
we give the predictions of the hydrodynamic theory on the studied time correlation functions. In section 3
we supply details of the MD simulations, and section 4 reports obtained the results on the autocorrelation
functions of energy, heat and entropy densities obtained bymolecular dynamics simulations of supercriti-
cal Ar, their corresponding time-Fourier transforms, and the estimated dependencies for the generalized
specific heats Cp(k) and Cv(k). The last section contains conclusions of this study.
2. Hydrodynamic approach
Hydrodynamic approach to collective dynamics of liquids is valid only on macroscopic length and
time scales where the most long-time relaxation processes in continuum are considered. The macro-
scopic equations in the hydrodynamic theory are in fact the local conservation laws for the densities of
particles, of total momentum and of energy. For the case of longitudinal dynamics, the only relevant on
macroscopic scale contributions to the hydrodynamic time correlation functions [8] are the ones coming
from acoustic longitudinal propagating modes with the linear in wave numbers k dispersion law
ωhyd(k) = csk
and quadratic in k damping
σhyd(k) = Γk2.
Here cs is the adiabatic speed of sound and the damping coefficient Γ= (DL+(γ−1)DT)/2 depends on the
longitudinal kinematic viscosity DL, the thermal diffusivity DT and the ratio of specific heats γ= Cp/Cv.
The only relevant relaxing mode for the longitudinal dynamics on macroscopic scales is the one coming
from thermal relaxation, i.e., connected with the diffusivity of local temperature DT.
13606-2
Heat capacity of liquids
The analytical expressions for the hydrodynamic time correlation functions, which we will study by
MD simulations, have in general a three term form: an exponential function connected with thermal
relaxing mode and two oscillating terms coming from the acoustic excitations [22]. The heat density au-
tocorrelation function reads
Fhh (k, t) = Ahh e−DTk2t + [Bhh cos cskt +Dhh (k)sin cskt ]e−Γk2t , (2.1)
where the amplitudes of mode contributions are constrained by the lowest-order sum rule
Ahh(k)+Bhh (k) = kBT 2Cv(k) . (2.2)
Note that the mode contributions Ahh and Bhh are constants in the long-wavelength limit, while the am-
plitide of the “non-Lorentzian” term Dhh (k) linearly depends on k. However, outside the hydrodynamic
region when one applies MD simulations the amplitudes Ahh and Bhh become k-dependent making pos-
sible estimation of the wave-number dependent specific heat. Similar is the expression for the energy
density autocorrelation function:
Fee (k, t) = Aee e−DTk2t + [Bee coscskt +Dee (k)sin cskt ]e−Γk2t , (2.3)
while for the entropy density autocorrelation function, the hydrodynamics predicts a single exponential
form for pure liquids:
Fss(k, t) = Asse−DTk2t , (2.4)
with
Ass(k) = kBCp(k) . (2.5)
Hence, the wavenumber-dependent specific heats Cv(k) and Cp(k) are connected with the static heat-
density and entropy-density autocorrelation functions, which in fact are the zeroth moments of the cor-
responding spectral functions Shh(k,ω) and Sss(k,ω). In the long-wavelength limit, the wavenumber de-
pendent specific heats tend to their macroscopic values Cv and Cp. The advantage of the hydrodynamic
approach lies in a possibility to estimate the contributions from relaxing and propagating modes to cor-
responding thermodynamic quantities [25].
3. Molecular dynamics simulations
Weperformedmolecular dynamics simulations for supercritical Ar at temperature T = 280 K and two
densities 750 kg/m3 and 1464.5 kg/m3 using a system of 2000 particles interacting via the Woon potential
[26] with the parameters taken from [27]. The cut-off radius for the effective two-body potential was 12 Å.
This potential permits a nice reproduction of the thermodynamic and dynamic quantities of supercritical
Ar by MD simulations as it was shown in [28].
The simulations were performed in the microcanonical ensemble with the time step of 2 fs. The du-
ration of production runs was of 360 000 time steps. Every sixth configuration was used for the sampling
of dynamic variables of particle density, momentum density, energy density and first time derivative of
the momentum density. The averages of static and time correlation functions over all possible directions
of different wave vectors with the same wave number were performed.
During the MD simulations we sampled the spatial Fourier components of hydrodynamic variables
of: particle density
n(k, t) =
1
p
N
N
∑
i=1
e−ikri (t ) , (3.1)
density of longitudinal momentum
J L(k, t) =
m
k
p
N
N
∑
i=1
(kvi)e−ikri(t) , (3.2)
13606-3
T. Bryk, T. Scopigno, G. Ruocco
and energy density
e(k, t) =
1
p
N
N
∑
i=1
εi (t)e−ikri (t ) , (3.3)
where ri (t), vi (t) and εi (t) are the trajectories, velocities and single-particle energies of the i -th particle
[14]. The single-particle energies were calculated as follows:
εi (t) =
mv2
i
(t)
2
+
1
2
∑
j,i
Φ
(
|ri (t)−r j (t)|
)
,
where Φ(r ) is the effective pair potential used in the simulations.
The above dynamic variables were used to calculate the hydrodynamic time correlation functions as
well as the heat density autocorrelation functions
Fhh (k, t) =
〈
h(k, t)h∗(k, t = 0)
〉
(3.4)
with
h(k, t) = e(k, t)−
fne
fnn
n(k, t), (3.5)
where the brackets mean the ensemble average and static averages read as follows fne (k) = 〈n(k)e(−k)〉
and fnn (k) = 〈n(k)n(−k)〉.
In order to study the entropy density autocorrelation functions Fss(k, t), we sampled the density of
the longitudinal component of stress tensor σL(k, t) via [14]
J̇ L(k, t) =−ikσL(k, t) ,
where the overdot denotes the time derivative of the corresponding dynamic variable. Hence, the dy-
namic variable of entropy density can be sampled from MD simulations as [14]
s(k, t) =
1
T
[
e(k, t)−
feσL
fσLn
n(k, t)
]
with the corresponding static averages feσL (k) = 〈e(k)σL(−k)〉 and fσLn (k) = 〈σL(k)n(−k)〉.
4. Results and discussion
Energy density autocorrelation functions Fee (k, t) = 〈e(k, t)e∗(k, t = 0)〉 calculated in MD simulations
are shown for three wavenumbers in small-k region in figure 1. The time dependence of these time cor-
relation functions is very similar to the relaxation of the heat density autocorrelations shown in figure 2.
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5
F
ee
(k
,t)
/(
k B
T
)2
t / τ
0.112 A-1(a)
0.194 A-1
0.317 A-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
F
ee
(k
,t)
/(
k B
T
)2
t / τ
0.140 A-1(b)
0.242 A-1
0.396 A-1
Figure 1. (Color online) Energy density autocorrelation functions Fee (k, t) at three wave numbers as
calculated from MD simulations for supercritical Ar at T = 280 K and densities 750 kg/m3 (a) and
1464.5 kg/m3 (b). The time scales τ are equal to 3.701 ps (a) and 2.961 ps (b).
13606-4
Heat capacity of liquids
A typical feature is an increasing amplitude and a decreasing frequency of the damped oscillations in
the shape of Fee (k, t) and Fhh (k, t) towards smaller wave numbers, which is typical of the contribution
coming from acoustic excitations. The relative strength of the relaxing and oscillating contributions in
Fhh (k, t) was estimated in the long-wavelength limit in [25] as the inverse of (γ−1), i.e., the inverse of
the Landau-Placzek ratio known for the density-density time correlation functions. Hence, for the case of
the two thermodynamic points of the supercritical Ar studied here and having the experimental ratio of
specific heats γ of 2.30 and 1.55 [24] for the low- and high-density studied systems, one should observe
much smaller relaxing contribution for the low-density state. Indeed, in figure 2 (a) the oscillations in
the shape of Fhh (k, t) even make the heat density autocorrelation function negative close to ωτ ∼ 1 for
the smallest possible in these simulations wave number, while in figure 2 (b), due to the much stronger
contribution from the relaxing thermal mode, the Fhh (k, t) is positive for all times. It is important that
the dimensionless functions Fhh (k, t) have the zero-time values tending to the macroscopic values of Cv
due to (2.2). Indeed, in figure 2 (a) the zero-time values tend to the macroscopic value of Cv = 1.775 [24],
while for the high-density state they tend to the value Cv = 2.34.
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5
F
hh
(k
,t)
/(
k B
T
)2
t / τ
0.112 A-1(a)
0.194 A-1
0.317 A-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
F
hh
(k
,t)
/(
k B
T
)2
t / τ
0.140 A-1(b)
0.242 A-1
0.396 A-1
Figure 2. (Color online) Heat density autocorrelation functions Fhh (k, t) at three wave numbers as calcu-
lated fromMD simulations for supercritical Ar at T = 280 K and densities 750 kg/m3 (a) and 1464.5 kg/m3
(b). The time scales τ are equal to 3.701 ps (a) and 2.961 ps (b).
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 1 2 3 4 5
F
ss
(k
,t)
/k
B
2
t / τ
0.112 A-1(a)
0.194 A-1
0.317 A-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3
F
ss
(k
,t)
/k
B
2
t / τ
0.140 A-1(b)
0.242 A-1
0.396 A-1
Figure 3. (Color online) Entropy density autocorrelation functions Fss (k, t) at three wave numbers as
calculated from MD simulations for supercritical Ar at T = 280 K and densities 750 kg/m3 (a) and
1464.5 kg/m3 (b). The time scales τ are equal to 3.701 ps (a) and 2.961 ps (b).
The entropy density autocorrelation functions Fss(k, t) calculated in MD simulations for two den-
sities of the supercritical Ar (figure 3) show for the smallest wave numbers a simple relaxation shape
without any oscillating contribution. This is in complete agreement with the prediction of the hydrody-
namic theory (2.4). The zero-time values of Fss(k, t) (2.5) should tend in the long-wavelength limit to the
13606-5
T. Bryk, T. Scopigno, G. Ruocco
macroscopic value of Cp. Indeed, one observes in figures 3 (a) and (b) that in agreement with this hydro-
dynamic prediction, the zero-time values of the normalized functions tend to the macroscopic values of
Cp reported in the NIST database: 4.097 for the low-density state and 3.62 for the high-density state.
The time-Fourier transformed energy density (e-e), heat energy density (h-h) and entropy density (s-s)
autocorrelation functions, Si i (k,ω), i = e,h, s are shown for the two studied densities of the supercritical
Ar in figure 4. The See (k,ω) and Shh (k,ω) have the typical of the dynamic structure factors shape with
the central peak due to thermal relaxation and the side peak due to propagating acoustic excitations.
The ratio of the integral intensities of the central and two side peaks known as the Landau-Placzek-
type ratio was obtained in [25] for the case of Shh(k,ω) and is equal to (γ− 1)−1. The side peak of the
See (k,ω) and Shh (k,ω) is the consequence of the corresponding oscillating contributions to the Fee (k, t)
and Fhh (k, t), respectively. It is remarkable that the calculated spectral function Sss(k,ω) shows only the
one-peak shape with the central peak due to thermal relaxation (line connected star symbols in figure 4).
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8
S
ii(
k,
ω
)
/ r
ed
.u
n.
ωτ
k=0.112 A-1
e-e(a)
h-h
s-s
0
1
2
3
4
5
6
7
0 2 4 6 8 10 12
S
ii(
k,
ω
)
/ r
ed
.u
n.
ωτ
k=0.140 A-1
e-e(b)
h-h
s-s
Figure 4. (Color online) Spectral functions energy-energy See (k,ω), heat-heat Shh(k,ω), entropy-entropy
Sss(k,ω) at the smallest available in MD simulations wave numbers for supercritical Ar at T = 280 K and
densities 750 kg/m3 (a) and 1464.5 kg/m3 (b). The time scales τ are equal to 3.701 ps (a) and 2.961 ps (b).
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
C
p(
k)
, C
v(
k)
k / A-1
Cv(k)
(a)
Cp(k)
MD
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4
C
p(
k)
, C
v(
k)
k / A-1
Cv(k)
Cp(k)
(b) MD
Figure 5. (Color online) Generalized wavenumber-dependent specific heats at constant volume Cv(k) and
constant pressure Cp(k) for supercritical Ar at T = 280 K ad densities 750 kg/m3 (a) and 1464.5 kg/m3 (b).
The asterisks at k = 0 correspond to macroscopic values of Cv calculated from the temperature fluctua-
tions [11].
The zero-time values of the time correlation functions Fhh (k, t) and Fss(k, t) permit estimation of the
wavenumber-dependent specific heats Cv(k) and Cp(k), which in the long-wavelength limit tend to their
macroscopic values. In figure 5 we show the wavenumber dependencies of Cv(k) and Cp(k) for the two
densities of the supercritical Ar at temperature 280 K. The asterisks at k = 0 show the values of macro-
13606-6
Heat capacity of liquids
scopic values of Cv obtained from the standard expression via the thermal fluctuations in the fluid system
[11]. One can see that the reported wave-number dependencies tend quite well to the experimental val-
ues for the supercritical Ar taken from the NIST database [24]:Cv=1.775kB and Cp=4.097kB for the density
of 750 kg/m3 and Cv=2.34kB and Cp=3.62kB for the density of 1464.5 kg/m3.
5. Conclusions
In the MD simulations we calculated the autocorrelation functions of energy density, heat density
and energy density and compared them for the smallest available wave numbers with the predictions
of the hydrodynamic theory. Our results give evidence of a nice correspondence of the hydrodynamic
description of time correlation functions and their zero-time values, which made it possible to estimate
the wavenumber-dependent specific heats Cv(k) andCp(k), with the MD-based estimation of their macro-
scopic values and experimental values for specific heats of supercritical Ar taken from the NIST database.
These results are very important from the viewpoint of analysing the contributions from thermal
relaxation and collective excitations to the specific heats of liquids. Our results and the hydrodynamic
approachmake evidence of the definitive role of the thermal relaxation in calculation of the specific heats
of fluids when it is estimated via dynamic processes. These results will urge a further understanding
of the dynamics of the supercritical state of matter, which shows many interesting features observed
recently in the scattering experiments and MD simulations on supercritical Argon [29, 30] and Oxygen
[31].
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http://dx.doi.org/10.1063/1.1674820
http://dx.doi.org/10.1016/j.fluid.2008.12.004
http://dx.doi.org/10.1016/j.supflu.2010.10.041
http://dx.doi.org/10.1063/1.3442412
http://dx.doi.org/10.1140/epjst/e2011-01419-x
http://dx.doi.org/10.1103/PhysRev.153.250
http://dx.doi.org/10.1088/0370-1328/88/1/318
http://dx.doi.org/10.1088/0034-4885/38/4/001
http://dx.doi.org/10.1103/PhysRevA.38.271
http://dx.doi.org/10.1080/00268979500100181
http://dx.doi.org/10.1103/PhysRevE.63.051202
http://dx.doi.org/10.1103/PhysRevE.56.2903
http://dx.doi.org/10.1080/00268976.2013.838313
http://dx.doi.org/10.1103/RevModPhys.38.205
http://dx.doi.org/10.1080/00319107108083815
http://dx.doi.org/10.1080/00319107408084276
http://webbook.nist.gov
http://dx.doi.org/10.1063/1.4774406
http://dx.doi.org/10.1016/0009-2614(93)85601-J
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Теплоємнiсть рiдин: Гiдродинамiчний пiдхiд
Т. Брик1,2, Т. Скопiньйо3, Дж. Руокко3,4
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, Україна
2 Iнститут прикладної математики та фундаментальних наук, Нацiональний Унiверситет “Львiвська
Полiтехнiка”, 79013 Львiв, Україна
3 Фiзичний факультет, Римський унiверситет “La Sapienza”, I–00185, Рим, Iталiя
4 Центр бiо-нано наук Sapienza, Iталiйський технологiчний iнститут, 295 вул. королеви Елени, I–00161,
Рим, Iталiя
Ми дослiджуємо автокореляцiйнi функцiї густин енергiї, тепла та ентропiї, обчислених при моделюван-
нi методом молекулярної динамiки для надкритичного Ar та порiвнюємо їх iз передбаченнями гiдроди-
намiчної теорiї. Показано, що передбачена гiдродинамiчною теорiєю одноекспонентна форма автоко-
реляцiйної функцiї густини ентропiї чудово вiдтворюється при малих хвильових числах молекулярною
динамiкою та дозволяє розрахунок залежної вiд хвильового числа питомої теплоємностi при сталому ти-
ску. Отриманi питомi теплоємностi, залежнi вiд хвильового числа, при сталому об’ємi та тиску, Cv (k) and
Cp (k), у довгохвильовiй границi добре узгоджуються з макроскопiчними експериментальними значен-
нями Cv and Cp для дослiджених термодинамiчних точок надкритичного Ar.
Ключовi слова: рiдини, термодинамiка, теплоємнiсть, гiдродинамiчна теорiя
13606-8
http://dx.doi.org/10.1063/1.1290131
http://dx.doi.org/10.1080/00268976.2011.617321
http://dx.doi.org/10.1038/nphys1683
http://dx.doi.org/10.1038/srep01203
http://dx.doi.org/10.1103/PhysRevLett.97.245702
Introduction
Hydrodynamic approach
Molecular dynamics simulations
Results and discussion
Conclusions
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