###### Abstract

One of the basic concepts of modern physics with a long prehistory is a fluid,
which means a substance that flows under an applied shear stress. In this
sense fluids form a wide subset of the phases of matter that includes liquids,
dense gases, plasmas, and to some extent even plastic solids. Fluidity is one
of the main dynamical characteristics that depends strongly on the details of
the local structure. And vice versa, dominant details of local structural
ordering in the arrangement of particles on some time scales are important for
understanding the dynamics of fluids. The orderliness over distances
comparable to the inter-particle distances is usually treated as the short-
range order, whereas the orderliness repeated over infinitely large distances
is called the long-range order. Both are absent in the ideal gas, but liquids
and amorphous solids exhibit the short-range order. The physics of crystalline
solids with the long-range order is well understood. In fluids, however, the
atomic structure is changing with time. How are specific features of
structural ordering reflected in the fluid dynamics? In this chapter we try to
find answers to some old and new questions that still make the fluid dynamics
still very attractive for the theoretical studies.

Index type ‘default’ not definedI can’t help

##  Chapter 0 Some Old And New Puzzles
In The Dynamics Of Fluids

###### Contents

  1. 0 Some Old And New Puzzles In The Dynamics Of Fluids
    1. 1 Introduction
    2. 2 What is a fluid?
    3. 3 Theoretical framework or brief introduction to statistical description
    4. 4 Generalized collective mode approach
    5. 5 Some problems related to the fluid dynamics
      1. 1 Single particle diffusion
      2. 2 Viscoelastic properties of fluids
      3. 3 Optical phonon-like excitations and fast sound
      4. 4 Dynamics of molten salts: optic-like excitations
      5. 5 Some rigorous relations and their applications
      6. 6 Dynamic crossover in supercritical fluids
    6. 6 Conclusions

###  1 Introduction

This Chapter is mainly based on the talk, delivered by one of us (I. M.) at
the Ising Lectures 2017. We do not deal directly with the critical phenomena,
but discuss some problems of the dynamics of fluids, where the terms order and
disorder play a significant role and cooperative phenomena are very important.
In this context our goal is rather close to the mainstream of Ising Lectures.
As stated in the book published recently in Ref. 1 on the occasion of the 20th
anniversary of the Ising Lectures:

> “Nowadays Ising-like models completed by ideas from complex system science
> come into play as simple models on the way to conceive the secrets of nature
> in all their complexity”.

In the same way we shall try to describe the fluid dynamics by relatively
simple models, which: i) are based on the microscopic foundations; ii)
whenever possible, yield rigorous analytical expressions for the quantities of
interest; iii) can be verified in the computer simulations. Realizing that
even the most refined model is nothing more but the “working tool” for the
researcher to explain the Nature’s mysteries, and that it could render an
eventual success only in a conjunction with a suitable theory (“there is
nothing as practical as a good theory”, as a well known aphorism states111It
is interesting that the origin of this aphorism was a subject of special
investigation[2].), we pay great attention to the selection of the theoretical
framework. We believe that the generalized collective mode (GCM) approach[3,
4, 5], which originates from the method of non-equilibrium statistical
operator (NSO) [6, 7, 8], is the very tool, being able to explain at least
some old puzzles in the fluid dynamics and to foresee or explain the new ones.

The topic we discuss in this Chapter has a long history but many problems are
still under discussion. We start in Sec. 2 from short motivation and
historical retrospectives, helping the readers to find themselves an answer to
the question “what is a fluid?” We want to note in advance that the answer is
ambiguous: the fluid can be gas-like, liquid-like, or even display the
properties of solids, depending on its preparation, external conditions, and
the experimental setup.

In Sec. 3 the main ideas of the NSO method are described. The transport
equations as well as the equations for the equilibrium time correlation
functions (TCFs) are derived for the case of small deviations from
equilibrium. On this basis we consider in more details the equations of
generalized hydrodynamics for a multi-component fluid and present expressions
for the generalized (dependent on the wavenumber $k$) thermodynamic quantities
and the generalized (dependent on the wavenumber $k$ and frequency $\omega$)
transport coefficients that can be obtained rigorously within the NSO method.
This theoretical framework opens new possibilities for the study of collective
dynamics, playing a significant role in fluids, and allows one to use the
concept of collective excitations based on the microscopic background. In Sec.
4 it is shown how such ideas could be realized within the GCM.

The applications of the theory, described in Sec. 3 and 4, to several physical
problems are presented in Sec. 5. Here we explore such phenomena as the
diffusion in one-component and binary fluids, the viscoelastic properties of
fluids under different conditions, the appearance and possibilities of
observation of the optical phonon-like excitations in binary and
multicomponent dense fluids, which sometimes manifest themselves as a fast
sound. In addition, the dynamics of molten salts with dominant Coulomb
interactions, leading to quite interesting phenomena, are considered. We study
also some useful outputs of the theory and discuss the rigorous relations for
the transport coefficients. In particular, it is shown that the so-called
“universal golden rule” for the partial conductivities in ionic liquids can be
simply derived in the generalized form. The nature of the dynamic crossover in
supercritical fluids is briefly discussed as well.

We conclude in Sec. 6 with some final remarks and a brief discussion of the
perspectives and some open questions.

###  2 What is a fluid?

The Encyclopaedia Britannica defines a fluid as

> “any liquid or gas or generally any material that cannot sustain a
> tangential, or shearing, force when at rest and that undergoes a continuous
> change in shape when subjected to such a stress. This continuous and
> irrecoverable change of position of one part of the material relative to
> another part when under shear stress constitutes flow, a characteristic
> property of fluids.”

There exists a special branch of physics that explores the flow properties of
fluids. Rheology studies the flow of matter, primarily in a liquid state, but
also as “soft solids” or solids under conditions, in which they react by
plastic flow rather than being deformed elastically in response to an applied
force. This term was coined by Eugene Bingham in 1920 from the suggestion made
by Markus Reiner, inspired by the aphorism attributed to Heraclitus,
“$\Pi\alpha\nu\tau\alpha$ $\rho\epsilon\iota$” that means “everything flows”
[9].

The ability of a fluid to flow is characterized mostly by its viscosity, which
is a measure of fluid’s resistance to gradual deformation by shear or tensile
stress. A fluid that has no resistance to shear stress is known as an ideal
one, and from the physics viewpoint is governed by the dissipationless Euler’s
equations [10]. All real fluids have a non-zero positive viscosity and are
described by the Navier–Stokes hydrodynamic equations [11, 12]. In such a
dissipative system the entropy is being produced, and the motion of the fluid
ceases after the external stress is removed.

A fluid with very high viscosity may appear as a solid. In this context, one
of the most curious experimental setups is worthy to be mentioned.

Figure 1: The University of Queensland pitch drop experiment, demonstrating
the viscosity of bitumen [13].

It is the so-called pitch drop experiment (see Fig. 1), used mainly for
educational purposes. The long-lasting measurement of the flow of a pitch
piece (a highly viscous fluid, a kind of bitumen) was started in 1927 by Prof.
Thomas Parnell of the University of Queensland in Brisbane, Australia, to
demonstrate to his students that some substances, which look like solids, are,
in fact, very-high-viscosity fluids. The eighth drop fell on November 28,
2000, allowing experimentalists to calculate that the pitch viscosity is
approximately 2.3$\times$1011 times greater than that of water. The 9-th drop
has separated from the funnel on April 24, 2014. In the same year, another
similar experiment, which began in 1914 and predated the Queensland’s one by
13 years, was reported in the media. However, the chosen kind of pitch is much
more viscous, and this experiment has not yet produced its first drop and is
not expected to be completed for at least 1000 years [14].

Elasticity is known as the ability of a body to resist a distorting influence
or a deforming force and to return to its original size and shape, when that
influence or the force is removed. In a solid, the shear stress is a function
of strain (Hooke’s law), but in a fluid, the shear stress is a function of the
strain rate (according to Pascal’s law). Unlike purely elastic substances,
fluids have both elastic and viscous features, and the interplay between them
depends on various factors.

Figure 2: Shear stress vs. strain rate dependence for various fluids.

Depending on the relationship between the shear stress and the rate of strain
and its derivatives, fluids are classified as:

  * •

Newtonian (the stress is directly proportional to the rate of strain);

  * •

Non-Newtonian (the stress depends on the rate of strain in a more complicated
manner, e.g. via its higher powers and derivatives).

From the presented in Fig. 2 schematic shear stress vs. strain rate diagram
for various kinds of fluids it follows that a non-Newtonian fluid behaves in a
solid-like manner at small strain rates (SR). At the moderate values of SR it
loses such features, while at large SR it resembles an ideal fluid (the
corresponding curve goes almost parallel to the horizontal axis).

We touch upon this classification of fluids in more detail in Sec. 5, when
speaking about the two-variable model for the shear dynamics of a simple
liquid. But the question of how the liquid can manifest its state within the
above classification scheme appears to have a very interesting demonstration
in the nature.

In Fig. 3, the snapshots of a basilisk lizards running on water are shown. The
Basiliscus (Basilisk) is a genus of large corytophanid lizards, which are
endemic to southern Mexico, Central America, and northern South America. They
are commonly known as the Jesus Christ lizards, or simply the Jesus lizards,
due to their ability to run on water as a biped for significant distances
before sinking. They can do this with the speed of 1.5 meters per second. The
running basilisk uses implicitly the solid-like behaviour of liquid when
slapping and stroking strongly and frequently on the water surface, whereas it
is recovering its position and sliding on the surface due to the viscosity of
an ordinary (Newtonian) liquid. The juvenile lizards can theoretically
generate a maximum total force more than twice their body weight, forcing the
ordinary water to behave so extraordinarily (like a non-Newtonian fluid).

Figure 3: Running on water: Three dimensional force generating by basilisk
lizards [15].

Unlike lizards, humans are not able to run on water. This conclusion stems
from the famous P. Kapitsa problem, to estimate the order of magnitude of the
speed with which a man must run so that he would not sink, posed by him in
1968 in the physics entrance exam. Rough estimations show that the water-
runner would need to strike downwards through the water at about 30 m/s, the
speed well beyond the human ability. Furthermore, to strike downwards at this
speed, he would require an average power output almost 15 times greater than
the maximum sustainable output for humans. An interesting study on this
subject has been reported in Ref. 16.

This problem can be approached in a more quantitative way, using the so-called
Deborah number222The Deborah number was originally proposed by Markus Reiner,
a professor at Technion in Israel, who chose the name inspired by a verse in
the Bible, stating “The mountains flowed before the Lord” in the song by the
prophet Deborah in the Book of Judges 5:5. in order to estimate the
“newtonianity” of fluid. The Deborah number, $\mbox{De}=t_{r}/t_{p}$, is
defined as the ratio of the time it takes for a material to adjust to applied
stresses or deformations (relaxation time $t_{r}$) and the characteristic time
scale $t_{p}$ of an experiment (or of a computer simulation) probing the
response of the material. It is often used in rheology to characterize the
fluidity of materials under specific flow conditions [17]. At lower Deborah
numbers, the material behaves in a more fluid-like manner, with an associated
Newtonian viscous flow. At higher Deborah numbers, the material behaviour
enters the non-Newtonian regime, being increasingly dominated by the
elasticity and demonstrating the solid-like features.

Figure 4: Viscoelasticity as a reflection of molecular rearrangement.

So the Basilisk lizards apparently “produce” sufficient Deborah numbers at the
surface layers of ordinary water, which in most cases is known to be a
Newtonian fluid with $\mbox{De}\ll 1$. To observe the non-Newtonian behaviour,
one has to achieve larger values of the Deborah numbers. This is possible with
increasing the relaxation time $t_{r}$ and decreasing the contact time
$t_{p}$. Larger relaxation times are typical for fluids that contain particles
with a complex macromolecular structure (see, for instance, Fig. 4). It can be
expected that the stage of the contact with the liquid surface is very
important, because of the opportunity to change the fluid structure locally in
the area of contact with the formation of mesoscopic inhomogeneities,
observed, in fact, experimentally [15]. This leads to an increase of the
effective relaxation time $t_{r}$. Thus, not only the time of contact with the
liquid, but also its influence on the liquid local mesostructure in the
contact area becomes important. There is a reason to believe that this last
factor should be taken into account in order to explain the ability of the
Basilisk lizards to run on the surface of water.

Looking at Fig. 4, one can ask another question: “Is it possible to observe
the non-Newtonian behaviour in Newtonian fluids? The obvious answer would be:
“Yes, but for the observation times much smaller than the relaxation time
$t_{r}$, which is close to the characteristic time of the short-range
ordering”.

Figure 5: The structure (upper panel) and the pair correlation functions
(lower panel) of a gas (a), liquid or glass (b), and a crystalline solid (c).

Liquids are known to posses an intermediate place between gases and solids
with respect to the short-range order. Unlike dilute gases (see Fig. 5a),
whose pair correlation functions have a single maximum at the distances
comparable to the particle diameter, fluids have the short-range order, which
manifests itself in several maxima of $g(r)$, located in the vicinity of the
1-st, 2-nd and 3-rd coordination spheres (see Fig. 5b). Crystalline solids, on
the other hand, are characterized by the long-ranged order, which is just
slightly perturbed by the thermal lattice vibrations (phonon modes), leading
to the well-pronounced extrema with a rather complicated structure (see Fig.
5c). Thus, the dominant structural ordering processes in a system depend
strongly on the phase considered.

It is important to note that the short-range ordering is typical for dense
fluids, just like for liquids. Moreover, the short-range ordering of
dynamically arrested particles is one of the characteristic features of
glasses. However, this is not the case for dilute gases, although their
symmetry properties are the same as in liquids and glasses. On the other hand,
the presence of the short-range order, when a certain particle falls into a
fairly stable environment of others, can be reflected in the dynamics as a
special form of motion of trapped particles (the cage effect), resembling the
phonon motion of particles in crystals. The physics of phonons in crystalline
solids with the long-range order is well understood. In liquids, however, the
atomic structure is changing with time, and the concept of phonons becomes
questionable for long observation times. Therefore, the question arises as to
how this process will manifest itself in the collective dynamics of dense
fluids, liquids, and glasses. It could give us a new tool to study the phase
states, using some dynamic dissimilarities in its collective dynamics.

A typical phase diagram of a simple fluid is presented in Fig. 6. The gas-
liquid critical point is shown by the circle. It is the end point of the
pressure–temperature curve that designates conditions under which the liquid
and the gas can coexist. The first theoretical description of the “gas–liquid”
phase transition, based upon a hypothesis regarding the form of the $p-V-T$
equation of state, was proposed by Van der Waals in his PhD Thesis [18].
Within this simplest theory the critical point can be easily determined.

Figure 6: Typical ($p,V,T$) phase diagram for a simple fluid.

Above the critical temperature and the critical pressure the region of a
supercritical fluid with some very specific features, being interesting for
practical applications, is located. An interesting concept of a supercritical
mesophase was proposed recently in Refs. 19, 20 that, being macroscopically
homogeneous but microscopically heterogeneous, exists above the critical point
and is bounded by weak higher-order percolation transitions, reflecting some
specific structural ordering of particles in a fluid. In the mesophase (see
Fig. 7) one can consider the fluid as a microscopically heterogeneous mixture
of gas-like and liquid-like particle clusters that at a certain density
suddenly begin to permeate the volume and become macroscopic. Such a new
viewpoint provides also the basis for further studies of the dynamical
properties and rheological behaviour of supercritical fluids.

Figure 7: The liquid-vapor coexistence line for argon from Ref. 19. The
available-volume (PA) and bonded-cluster (PB) percolation transitions define
the bounds of a supercritical mesophase.

###  3 Theoretical framework or brief introduction to statistical description

Basic requirements for the microscopic theory, capable of describing the
statistical non-equilibrium thermodynamics and the dynamic properties of
fluids, in terms of current knowledge would look as follows:

  * •

it has to describe processes on different time and spatial scales;

  * •

it has to be flexible and convenient for application as well as for
interpretation of the results;

  * •

it should be formulated in the computer-adaptable form;

  * •

it has to ensure correct thermodynamic and hydrodynamic limits and allow a
transparent generalization to a wide range of wave vectors and frequencies.

  * •

one should have control over approximations (analogue of the sum rules) within
the elaborated theory.

It is clear that the path to creating such a theory was very long and
difficult. Although attempts to formulate certain aspects of the kinetic
theory of fluids were made already by Euler, Bernoulli, and Clausius [21], its
modern form is based on the contributions of James Clerk Maxwell (1831-1879),
Ludwig Boltzmann (1844-1906), and their followers [22, 23]. Since the
structure of liquids is much more complicated than that of gases, even
numerous generalizations of the Boltzmann kinetic equation failed to describe
the liquid dynamics due to the necessity to take the cooperative phenomena
into account (the analogue of many-particle collision in the kinetic theory).
Less than one century ago, the scientists even questioned the very ability of
the statistical physics to describe the liquid state of matter. In 1937, to
mark the centenary of Van der Waals, the conference on statistical mechanics
was held in Amsterdam, attended by famous scientists such as P. Debye, G.
Ulenbech, and many others. The session chairman H. Kramers put the issue
“whether the statistical mechanics can describe the liquid region” to the
vote. An outcome turned out to be a draw: 50 to 50, with P. Debye voting
“Nay”. Now we know that the correct answer should have been “Yea”. But today
we may vote “whether the statistical theory can describe the dynamics of
complex fluids”.

Skipping the mid-20th century works by M. Born, H. S. Green and J. G.
Kirkwood, who had greatly contributed to creation of the kinetic theory of
fluids, we have to mention M. M. Bogolyubov333A more “traditional” spelling of
the name of this great mathematician and physicist of the XX-th century is N.
N. Bogoliubov, which is a transliteration from the Russian version of the
name, Nikolai Nikolajevich Bogoliubov, see Ref. 24. However, here we use the
transliteration from Ukrainian, Mykola Mykolajovych Bogolyubov. The close ties
of M. M. Bogolyubov with Ukraine were studied in detail by the authors of this
Chapter and the Editor of this volume in Ref. 25. (the main “co-author” of the
BBGKI hierarchy concept) [24], whose ideas of the abbreviated description of
the system dynamics were creatively developed and extended by D. N. Zubarev.

Figure 8: Dmitry Zubarev (November 27, 1917 – July 29, 1992), known for his
contributions to statistical mechanics, non-equilibrium thermodynamics, plasma
physics, theory of turbulence, and to the development of the two-time Green’s
function formalism.

The main ideas of his method of NSO [6, 7, 8] are presented below.

Zubarev proposed to start from the time-reversible Liouville equation (LE) for
the statistical operator $\rho(\mathbf{x}^{N};t)$, depending on the phase
space $\mathbf{x}^{N}=\\{\mathbf{p,r}\\}^{N}$ and time $t$, and, using the
Bogolyubov’s ideas of the weakening of correlations, to average the formal
solution of the LE

$\rho(\mathbf{x}^{N};t)=\exp\\{-i\hat{L}(t-t^{\prime})\\}\rho_{q}(\mathbf{x}^{N};t^{\prime})$
(1)

(hereafter $\hat{L}$ means the Liouville operator) with the initial condition
$\rho(\mathbf{x}^{N};t^{\prime})=\rho_{q}(\mathbf{x}^{N};t^{\prime})$ over the
time interval $t-t_{0}$,

$\rho(\mathbf{x}^{N};t)=\frac{1}{t-t_{0}}\int\limits_{t_{0}}^{t}e^{-i\hat{L}(t-t^{\prime})}\rho_{q}(\mathbf{x}^{N};t^{\prime})dt^{\prime}.$
(2)

This interval has to be large enough (more precisely, $t-t_{0}\to\infty$) to
ensure initial states’ damping and formation of the dynamical correlations.
Using the Abel’s theorem [6, 7], one can obtain the LE

$\left(\frac{\partial}{\partial
t}+i\hat{L}\right)\rho(\mathbf{x}^{N};t)=-\epsilon\\{\rho(\mathbf{x}^{N};t)-\rho_{q}(\mathbf{x}^{N};t)\\}$
(3)

with the infinitesimal source $\epsilon$, which violates the time
reversibility and tends to zero, $\epsilon\to 0$, after performing the
thermodynamic limit $N\to\infty$, $V\to\infty$, $N/V=const$. The second
essential point of the NSO construction, proposed by Zubarev, is a selection
of the so-called quasi-equilibrium (or relevant) statistical operator
$\rho_{q}(\mathbf{x}^{N};t)$ in the generalized Gibbs-like form

$\rho_{q}(\mathbf{x}^{N};t)=\exp\left\\{-\Phi(t)-\sum\limits_{\alpha}\hat{P}_{\alpha}F_{\alpha}(t)\right\\}$
(4)

in terms of the densities $\hat{P}_{\alpha}$ of slowly varying dynamical
variables of the abbreviated description along with the corresponding
conjugate thermodynamic forces $F_{\alpha}(t)$. The first term, $\Phi(t)$, in
the exponent of Eq. (4) denotes the Massier-Planck functional which insures
normalisation of the quasi-equilibrium statistical operator to unity. Since
the densities of the dynamic variables describe the local properties of the
system, they depend on the space coordinate $\mathbf{r}$ (or wavevector
$\mathbf{k}$ in the Fourier space). Hence, in the above and following
expressions an integration over the space coordinate has to be performed
explicitly along with the summation over the $\alpha$ index of dynamic
variables.

One can obtain the formal solution (3) for the NSO

$\rho(\mathbf{x}^{N};t)=\rho_{q}(\mathbf{x}^{N};t)-\\!\\!\int\limits_{-\infty}^{t}e^{-\epsilon(t-t^{\prime})}T(t,t^{\prime})(1-{\cal
P}_{q}(t^{\prime}))i\hat{L}t^{\prime}\rho_{q}(\mathbf{x}^{N};t^{\prime})dt^{\prime}$
(5)

as the functional of observables
$\mbox{Sp}\\{\hat{P}_{\alpha}\rho(\mathbf{x}^{N};t)\\}\equiv\langle\hat{P}_{\alpha}\rangle^{t}=\langle\hat{P}_{\alpha}\rangle^{t}_{q}$,
where $\mbox{Sp}\\{\cdots\\}$ means an integration over the phase space for
the classical systems (or taking the corresponding trace for the quantum
ones), and $\langle\ldots\rangle_{q}^{t}$ means averaging with the quasi-
equilibrium statistical operator $\rho_{q}(\mathbf{x}^{N};t)$. In Eq. (5)

$T(t,t^{\prime})=\exp_{+}\left\\{-\int\limits_{t^{\prime}}^{t}d\tau(1-{\cal
P}_{q}(\tau))i\hat{L}\right\\}$

denotes the chronologically ordered exponent, constructed on the Kawasaki-
Gunton projection operators ${\cal P}_{q}(t)$, which act in the space of
statistical operators [6, 26].

Using (4)-(5), one can write down the dynamic equations for observables,

$\frac{\partial}{\partial t}\langle\hat{P}_{\alpha}\rangle^{t}=\langle
i\hat{L}P_{\alpha}\rangle^{t}_{q}+\sum\limits_{\beta}\int\limits_{-\infty}^{t}dt^{\prime}e^{-\epsilon(t-t^{\prime})}\left(\hat{I}_{\alpha}(t),T(t,t^{\prime})\hat{I}_{\beta}(t^{\prime})\right)_{q}F_{\beta}(t^{\prime}),$
(6)

where

$\displaystyle\left(\hat{I}_{\alpha}(t),T(t,t^{\prime})\hat{I}_{\beta}(t^{\prime})\right)_{q}$
$\displaystyle=\int\limits_{0}^{1}d\tau\langle\hat{I}_{\alpha}(t),[\rho_{q}(\mathbf{x}^{N};t^{\prime})]^{\tau}T(t,t^{\prime})\hat{I}_{\beta}(t^{\prime})[\rho_{q}(\mathbf{x}^{N};t^{\prime})]^{-\tau}\rangle_{q}^{t}$
(7)

denotes the correlation functions, constructed on the generalized fluxes

$\hat{I}_{\alpha}(t)=(1-{\cal P}(t))i\hat{L}\hat{P}_{\alpha},$ (8)

which are defined by the Mori projection operators ${\cal P}(t)$, acting in
the space of dynamical variables [26].

Assuming the deviation of the generalized thermodynamic forces $\delta
F_{n}(t)=F_{n}(t)-F_{n}^{0}$ from their equilibrium values $F_{n}^{0}$ to be
small, one can exclude them from expression (6) for the observables. In the
weakly non-equilibrium case, it is useful to rewrite the transport equation
(6) in terms of the Fourier transforms
$\langle\Delta\hat{P}_{\alpha}\rangle^{\omega}$ of fluctuations of the
dynamical variables
$\Delta\hat{P}_{\alpha}=\hat{P}_{\alpha}-\langle\hat{P}_{\alpha}\rangle_{0}$
around their equilibrium values $\langle\hat{P}_{\alpha}\rangle_{0}$ as
follows:

$\left\\{i\omega{\cal
I}-i\Omega_{0}+\tilde{\varphi}_{\epsilon}(\omega)\right\\}\langle\Delta\hat{P}_{\alpha}\rangle^{\omega}=0.$
(9)

The second term in (9) denotes the so-called frequency matrix

$i\Omega=\left(i\hat{L}\hat{P},\Delta\hat{P}\right)\left(\Delta\hat{P},\Delta\hat{P}^{+}\right)^{-1},$
(10)

which describes the non-dissipative processes in the system and is expressed
via the corresponding equilibrium correlation functions; ${\cal I}$ means the
identity matrix, whose dimension is equal to the number of dynamical variables
$\langle\Delta\hat{P}_{\alpha}\rangle^{\omega}$ considered. The third term in
Eqn. (9),

$\tilde{\varphi}(\omega)\\!=\\!\\!\left(\\!(1-{\cal
P})i\hat{L}\hat{P},\frac{1}{i\omega+\epsilon+(1-{\cal P})i\hat{L}}(1-{\cal
P})i\hat{L}\hat{P}^{+}\\!\right)\\!\left(\Delta\hat{P},\Delta\hat{P}^{+}\right)^{-1}$
(11)

denotes the matrix of memory functions, which determines the dissipation
processes in the system.

Since (9) represents the system of homogenous linear equations, the
determinant of the matrix in the curly brackets has to be equal to zero,
defining thereby the spectrum of collective modes. We touch upon this issue in
much more detail in Sec. 4, when speaking about the GCM approach. The equation
for Laplace transforms
$\left(\Delta\hat{P},\Delta\hat{P}^{+}\right)^{z}\equiv$
$\left(\Delta\hat{P},[z+i\hat{L}]^{-1}\Delta\hat{P}^{+}\right)$ of the
corresponding TCFs reads in a similar way

$\Big{\\{}i\omega{\cal
I}-i\Omega_{0}+\tilde{\varphi}(z)\Big{\\}}\left(\Delta\hat{P},\Delta\hat{P}^{+}\right)^{z}=\left(\Delta\hat{P},\Delta\hat{P}^{+}\right),$
(12)

with the r.h.s. of Eq. (12) being the static correlation functions (SCFs),
constructed on fluctuations of the dynamic variables. It should be stressed
that the matrix equation (12) for the Laplace transforms of TCFs is, in fact,
an identity. This can be easily proved using the expressions for the frequency
matrix (10) and the matrix of memory functions (11).

Before proceeding to the description of the GCM approach, it would be
instructive to look at the definitions for generalized thermodynamic
quantities and generalized transport coefficients that follow rigorously from
the expressions given above. For this purpose, let us consider a
$\nu$-component mixture and choose the densities of conserved variables as the
basic set of dynamic variables
$\hat{P}_{\mathbf{k}}=\left\\{\\{\hat{n}_{\mathbf{k},a}\\},\hat{\mathbf{J}}_{\mathbf{k}},\hat{H}_{\mathbf{k}}\right\\}$,
where

$\hat{n}_{\mathbf{k},a}=\sum\limits_{i=1}^{N_{a}}\exp\\{i\mathbf{k}\mathbf{R}_{i}^{a}\\}$
(13)

denotes the number density of particles of the $a$-th species;

$\hat{\mathbf{J}}_{\mathbf{k}}=\sum\limits_{a}\sum\limits_{i=1}^{N_{a}}\mathbf{p}_{i}^{a}\exp\\{i\mathbf{k}\mathbf{R}_{i}^{a}\\}$
(14)

is the density of the total current;

$\hat{H}_{\mathbf{k}}=\hat{E}_{\mathbf{k}}-\sum\limits_{ab}(\hat{E}_{\mathbf{k}},\hat{n}_{-\mathbf{k},a})(\hat{n}_{\mathbf{k}},\hat{n}_{-\mathbf{k}})^{-1}_{ab}\hat{n}_{\mathbf{k},b}$
(15)

is the enthalpy density, obtained by orthogonalization of the total energy
density

$\hat{E}_{\mathbf{k}}=\sum\limits_{a}\sum\limits_{i=1}^{N_{a}}e_{i}^{a}\exp\\{i\mathbf{k}\mathbf{R}_{i}^{a}\\},\quad
e_{i}^{a}=\frac{(\hat{\mathbf{p}}_{i}^{a})^{2}}{2m_{a}}+\frac{1}{2}\sum\limits_{b}\sum\limits_{j=1}^{N_{b}}V(|\mathbf{R}_{i}^{a}-\mathbf{R}_{j}^{b}|).$
(16)

In (13)-(16) the summation over $a,b$ runs from 1 up to the number of species
$\nu$; $\mathbf{k}$ denotes the wavevector; $\mathbf{R}_{i}^{a}$ and
$\mathbf{p}_{i}^{a}$ mean the particle position and momentum, respectively;
$m_{a}$ is the mass of the particle of the $a$-th species, while
$V(|\mathbf{R}_{i}^{a}-\mathbf{R}_{j}^{b}|)$ means the pairwise interaction
potential.

It was shown in Ref. 4 that the generalized ($k$-dependent) thermodynamic
quantities can be defined via corresponding SCFs, constructed on the densities
$\hat{P}_{\mathbf{k}}$. For instance, the isothermal compressibility
$\kappa_{T}(k)$ can be written as follows

$\frac{1}{\kappa_{T}(k)}=Nnk_{B}T\sum\limits_{ab}c_{a}(\hat{n}_{\mathbf{k}},\hat{n}_{-\mathbf{k}})^{-1}_{ab}c_{b},$
(17)

where $N$ denotes the total particle number, $n=N/V$, $V$ means the system
volume, $c_{a}=N_{a}/N$ is the concentration of the $a$-th species and $k_{B}$
is the Bolzmann constant. The specific heat at constant volume $C_{V}(k)$ is
defined by

$k_{B}T^{2}C_{V}(k)=(\hat{H}_{\mathbf{k}},\hat{H}_{-\mathbf{k}}).$ (18)

The linear thermal expansion coefficient $\alpha_{P}(k)$ reads

$\alpha_{P}(k)=\frac{1}{ik}(i\hat{L}\hat{J}_{\mathbf{k}}^{||},\hat{H}_{-\mathbf{k}})\frac{\kappa_{T}(k)}{k_{B}T^{2}V},$
(19)

where $\hat{J}_{\mathbf{k}}^{||}$ denotes the longitudinal component of the
momentum density. And for the generalized ratio $\gamma(k)=C_{P}(k)/C_{V}(k)$
the relation

$\gamma(k)=1+\frac{TV\alpha_{P}^{2}(k)}{C_{V}(k)\kappa_{T}(k)},$ (20)

being well-known from the standard thermodynamics, is satisfied.

In a similar way, the elements of memory functions matrix (11) can be related
to the generalized transport coefficients [4], for example:

$\tilde{\varphi}_{nn}^{ab}(k,z)=k^{2}N\sum\limits_{c}D_{ac}(k,z)(\hat{N},\hat{N}^{+})^{-1}_{cb},$
(21)

where $D_{ab}(k,z)$ are the generalized mutual diffusion coefficients;

$\displaystyle\tilde{\varphi}_{nh}^{a}(k,z)=\frac{k^{2}}{TC_{V}(k)}D_{T}^{a}(k,z),$
(22)
$\displaystyle\tilde{\varphi}_{hn}^{a}(k,z)=k^{2}Vk_{B}T\sum\limits_{b}D_{T}^{b}(k,z)(\hat{N},\hat{N}^{+})^{-1}_{ba},$

with $D_{T}^{a}(k,z)$ being the generalized thermal diffusion coefficients;

$\tilde{\varphi}_{JJ}(k,z)=\frac{k^{2}}{\rho}\eta_{l}(k,z)$ (23)

is connected with the generalized longitudinal viscosity
$\eta_{l}(k,z)=4/3\eta(k,z)+\zeta(k,z)$, where $\eta(k,z)$ and $\zeta(k,z)$
denote, respectively, the generalized shear and bulk viscosities, and,
finally,

$\tilde{\varphi}_{hh}(k,z)=\frac{k^{2}V}{C_{V}(k)}\lambda(k,z),$ (24)

where $\lambda(k,z)$ is the generalized thermal conductivity. There are also
the generalized transport coefficients $\xi(k,z)$ and $\zeta_{a}(k,z)$,

$\displaystyle\tilde{\varphi}_{jh}(k,z)=-\frac{ik^{2}}{TC_{V}(k)}\xi(k,z),$
$\displaystyle\tilde{\varphi}_{hj}(k,z)=-\frac{ik^{2}}{\rho}\xi(k,z),$ (25)
$\displaystyle\tilde{\varphi}_{nj}^{a}(k,z)=-\frac{ik^{2}}{\rho}\zeta_{a}(k,z),$
$\displaystyle\tilde{\varphi}_{jn}^{a}(k,z)=-ik^{2}TV\sum\limits_{b}\zeta_{b}(k,z)(\hat{N},\hat{N}^{+})^{1}_{ba},$

which describe the heat–viscosity and diffusion–viscosity dynamic cross-
correlations, respectively. Since the corresponding memory functions are
constructed on the fluxes of different tensor dimensionality, they contribute
in higher order in the wavenumber $k$ and, therefore, can be neglected in the
hydrodynamic limit $k\to 0$.

To summarize this Section, let us emphasize that Eqs. (9) and (12) in their
present form do not allow one to explore the fluid dynamics, since the memory
functions cannot be calculated exactly, and some approximation for
$\tilde{\varphi}(\omega)$ is necessary. One of the most efficient ways to
proceed is to use the Markovian approximation
$\tilde{\varphi}(z)\approx\tilde{\varphi}(0)$ for the memory kernels, which
from the physical point of view means that all the dissipative processes in
the system are assumed to decay in time very fast. Such an approximation is
attractive also because it makes it possible to use computer simulations for
the alternative calculations of $\tilde{\varphi}(0)$. However, for many
applications it is important to take into account the $\omega$-dependence of
the memory effects. It is possible to do by extending the set of dynamical
variables by taking higher derivatives
$(i\hat{L})^{\alpha}\hat{P}_{\mathbf{k}}$, $\alpha>1$, into account, where
$\hat{P}_{\mathbf{k}}$ denote the densities of the conserved quantities. This
idea, which is the basis of the GCM method, rests upon quite a reasonable
assumption that the relaxation times of the kernels, built on the extended set
of dynamic variables, are much shorter than those dealt with the basic
variables $\hat{P}_{\mathbf{k}}$ alone. In such a case the Markovian
approximation for higher order memory functions is well justified, and it is
possible to obtain the closed form of equations (12) in terms of the SCFs and
relaxation times for TCFs of the hydrodynamic origin. Some mathematical
aspects of solving Eqs. (12) are considered in the next Section, along with an
analysis of the applicability of the GCM approach from the point of view of
the “sum rules” requirement.

###  4 Generalized collective mode approach

Before introducing the basic ideas of the GCM approach, we would like to
discuss briefly the experimental motivation for such a study and the current
status of the theory in this domain. To begin with, let us explain the
physical meaning of the “collective modes in fluids” term. The collective
modes (or collective excitations) of a certain physical nature, formally
decoupled from each other, 444Frequently, such a mode decoupling exists only
in a certain domain of the wavenumbers $k$, and there could be an essential
overlap of collective excitations at other $k$, which is observed in the shape
of the dynamic structure factor or “momentum-momentum” spectral function,
making it difficult to relate unambiguously a particular mode to certain
physical phenomenon. It is an advantage of the GCM approach that allows to
solve such puzzles efficiently in many cases. can be observed, directly or
indirectly, in scattering experiments, measurements of the sound velocity and
attenuation, structural relaxation in glasses, peculiarities of the
viscoelastic behaviour of fluids, and so on. These modes can be of propagating
nature, if they correspond to the sound [5, 27], shear [28], heat [29], charge
[30], or magnetic (spin) [31] waves in fluids. They can be also purely
relaxing [5], when they describe diffusion-like or relaxational processes.
Depending on their behaviour at small wavenumbers $k$, they can be divided
into the hydrodynamic or kinetic subcategories. The hydrodynamic ones are
connected with the slowest processes in the hydrodynamic regime, which reflect
the dynamics of the conserved quantities and determine the main features of
the Navier-Stokes hydrodynamics. Its damping coefficients tend to zero in the
$k\to 0$ limit. Otherwise, the kinetic-like modes tend to non-zero values at
$k\to 0$, reflecting the processes of kinetic origin.

In the middle of the XX century, the so-called shear waves were identified in
computer simulations. In particular, it has been found that the shape of the
transverse spectral function is changed [32], when the wavenumber $k$ exceeds
a certain value, typical for each particular liquid. This means that the
viscoelastic properties of liquids for smaller spatial scales become similar
to those of solids, and liquids behave like elastic bodies. Similar behaviour
was observed later for the thermal properties, and this allowed one to
identify the so-called heat waves [33]. In the second part of the 1980ies, the
so-called fast sound modes were found [34] in the scattering experiments,
showing the crossover between different types of collective behaviour in
dynamics of binary mixtures. About ten years before that, Hansen and McDonald
[35] carried out a pioneering molecular dynamics (MD) simulation for a model
of a molten salt and observed the propagating optic-like modes in the “charge-
charge” dynamical structural factors. All these excitations are the examples
of propagating modes that could not be described within the standard
hydrodynamic theory, so that a lot of different kinds of theories were
proposed in order to explain such phenomena [32, 36, 37]. However, their main
disadvantage was that these theories were formulated mainly for needs of some
specific experiments. In addition to the non-hydrodynamic (kinetic)
propagating modes, relaxing kinetic-like collective modes were identified in
the experiments, that could not be explained within the hydrodynamic theory.
For instance, we can mention the structural relaxation [38], playing a crucial
role in the glass-forming liquids, and the molecular relaxation [39],
reflecting the contribution of additional degrees of freedom that are
important in complex fluids for certain domains of wavenumbers $k$ and
frequencies $\omega$.

A big challenge for a theory is dealt with the collective dynamics of the
multi-component mixtures. In this case, even the problems of hydrodynamic
theory have not been completely solved. In particular, no analytical solution
is found for the hydrodynamic TCFs of a mixture with more than three species,
indispensable for the correct interpretation of scattering experiments. The
question of correct expressions for the hydrodynamic fluxes, that have to be
used for calculations of transport coefficients in the multicomponent fluids,
is still discussed in literature since this problem is important for many
practical applications. Hence we may conclude that there is still a need in
the generalized hydrodynamic theories giving us an opportunity to consider all
these different phenomena within a unified scheme.

To formalize the main postulates of the GCM approach, let us consider the
extended set of dynamic variables
$\hat{\textbf{P}}^{(s)}=\left\\{\hat{\textbf{P}}_{0},\hat{\textbf{P}}_{1},\ldots,\hat{\textbf{P}}_{s}\right\\}$,
$\hat{\textbf{P}}_{s}=(i\hat{L})^{s}\hat{\textbf{P}}_{0}$, where
$\hat{\textbf{P}}_{0}$ denotes the densities of hydrodynamic variables (for
instance, the values (13)-(15) in the case of a multi-component fluid). If one
passes to the orthogonalized set of the dynamic variables according to the
well-known Gramm-Schmidt procedure,

$\displaystyle\hat{\mathbf{Y}}_{0}=\hat{\mathbf{P}}_{0},\quad\hat{\mathbf{Y}}_{1}=(1-{\cal
P}_{0})\hat{\mathbf{P}}_{1},\ldots,\hat{\mathbf{Y}}_{s}=(1-{\cal
P}_{s-1})\hat{\mathbf{P}}_{s},$ (26)

where the projection operators ${\cal P}_{s}$ are defined in the standard way,

$\displaystyle{\cal P}_{0}=\Delta{\cal
P}_{0}=(\ldots,\hat{\mathbf{P}}_{0}^{+})(\hat{\mathbf{P}}_{0},\hat{\mathbf{P}}_{0}^{+})^{-1}\hat{\mathbf{P}}_{0},$
$\displaystyle{\cal P}_{s}=\sum\limits_{s^{\prime}=0}^{s}\Delta{\cal
P}_{s^{\prime}},\qquad\Delta{\cal
P}_{s^{\prime}}=(\ldots,\hat{\mathbf{Y}}_{s^{\prime}}^{+})(\hat{\mathbf{Y}}_{s^{\prime}},\hat{\mathbf{Y}}_{s^{\prime}}^{+})^{-1}\hat{\mathbf{Y}}_{s^{\prime}},$

one can generalize the transport equations (9) as follows:

$\left\\{i\omega{\cal
I}-i\mathbf{\Omega}^{(s)}+\tilde{\mathbf{\varphi}}^{(s)}(\omega)\right\\}\langle\Delta\hat{\mathbf{Y}}_{s}\rangle^{\omega}=0.$
(28)

The frequency matrix $i\mathbf{\Omega}^{(s)}$ in Eq. (28) is of the three-
diagonal structure,

$i\mathbf{\Omega}^{(s)}=\left(\begin{array}[]{cccccc}i\mathbf{\Omega}_{0}&{\cal
I}&&&&\\\ -\mathbf{\Gamma}_{0}&i\mathbf{\Omega}_{1}&{\cal I}&&0\\\
&-\mathbf{\Gamma}_{1}&i\mathbf{\Omega}_{2}&{\cal I}&&\\\
&&\ldots&\ldots&\ldots&\\\
&0&&-\mathbf{\Gamma}_{s-2}&i\mathbf{\Omega}_{s-1}&{\cal I}\\\
&&&&-\mathbf{\Gamma}_{s-1}&i\mathbf{\Omega}_{s}\end{array}\right),$ (29)

and its components are equal to

$\displaystyle
i\mathbf{\Omega}_{l}=i\mathbf{\Omega}_{ll}=(i\hat{L}\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})(\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})^{-1},$
$\displaystyle
i\mathbf{\Omega}_{l-1l}=(i\hat{L}\hat{\mathbf{Y}}_{l-1},\hat{\mathbf{Y}}_{l}^{+})(\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})^{-1}=(\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})(\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})^{-1}\equiv{\cal
I}$
$\displaystyle\mathbf{\Gamma}_{l}=-i\mathbf{\Omega}_{l+1l}=-(i\hat{L}\hat{\mathbf{Y}}_{l+1},\hat{\mathbf{Y}}_{l}^{+})(\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})^{-1}=(\hat{\mathbf{Y}}_{l+1},\hat{\mathbf{Y}}_{l+1}^{+})(\hat{\mathbf{Y}}_{l},\hat{\mathbf{Y}}_{l}^{+})^{-1}.$

The matrix of memory functions $\tilde{\varphi}^{(s)}$ in Eq. (28),

$\tilde{\varphi}^{(s)}=\left(\begin{array}[]{ccccc}0&0&\ldots 0&0\\\
0&0&\ldots 0&0\\\ &&\ldots&&\\\ 0&0&\ldots 0&0\\\ 0&0&\ldots
0&\tilde{\varphi}_{s}\\\ \end{array}\right),$ (31)

has only one non-zero block,

$\tilde{\varphi}_{s}(z)\\!=\\!\\!\left(\\!\hat{\mathbf{Y}}_{s+1},\frac{1}{z+(1-{\cal
P})i\hat{L}}\hat{\mathbf{Y}}_{s+1}^{+}\\!\right)\\!\left(\hat{\mathbf{Y}}_{s},\hat{\mathbf{Y}}_{s}^{+}\right)^{-1}.$
(32)

The system of transport equations (28) can be solved starting from the
variables $\langle\Delta\hat{\mathbf{Y}}_{s}\rangle^{\omega}$ up to
$\langle\Delta\hat{\mathbf{Y}}_{s-1}\rangle^{\omega}$ (the so-called rolling
up procedure). Thus the lower-order memory function $\tilde{\varphi}_{s-1}(z)$
can be related to the higher-order ones $\tilde{\varphi}_{s}(z)$ via the
recurrent matrix expression

$\tilde{\varphi}_{s-1}(z)=\left[z{\cal
I}-i\mathbf{\Omega}_{s}+\tilde{\varphi}_{s}(z)\right]^{-1}\mathbf{\Gamma}_{s-1}.$
(33)

Performing the rolling up procedure to the lowest order dynamical variables,
one can attribute the obtained expressions at
$\langle\Delta\mathbf{Y}_{0}\rangle^{\omega}$ to the generalized ($k$ and
$\omega$ dependent) transport coefficients [40].

The generalization of the dynamic equations for the TCFs constructed on the
orthogonalized basic set (26) is straightforward (cf. Eq. (12)). At this
stage, it is well justified to perform the Markovian approximation for the
memory function $\tilde{\varphi}_{s}(z)$. Some general physical reasons to do
so have been already formulated at the beginning of this Section. Moreover,
there are even stricter ideas based on the so-called “non-Markovianity
spectrum” elaborated in Refs. 41, 42. We will not go into details of this
issue here, but return to it at the analysis of the obtained results.

In the Markovian approximation, the equations for TCFs look as follows:

$\displaystyle\left[z{\cal
I}+\mathbf{T}^{(s)}(k)\right]\tilde{\mathbf{F}}^{(m,s)}(k,z)=\mathbf{F}^{(s)}(k),$
(34)
$\displaystyle\tilde{\mathbf{F}}^{(m,s)}(k,z)\\!=\\!\left(\\!\Delta\hat{\mathbf{Y}_{\mathbf{k}}}^{(s)},[\Delta\hat{\mathbf{Y}_{-\mathbf{k}}}^{(s)}]^{+}\\!\right)^{z},\,\tilde{\mathbf{F}}^{(s)}(k)\\!=\\!\left(\\!\Delta\hat{\mathbf{Y}_{\mathbf{k}}}^{(s)},[\Delta\hat{\mathbf{Y}_{-\mathbf{k}}}^{(s)}]^{+}\\!\right),$

where

$\mathbf{T}^{(s)}(k)=i\mathbf{\Omega}^{(s)}(k)+\tilde{\varphi}^{(s)}(k,0)$
(35)

denotes the so-called generalized hydrodynamic matrix [3, 4]. In Eq. (34) we
denote the basic set of dynamic variables as
$\hat{\mathbf{Y}}^{(s)}=\\{\hat{\mathbf{Y}}_{0\mathbf{k}},\hat{\mathbf{Y}}_{1\mathbf{k}},\ldots,\hat{\mathbf{Y}}_{s\mathbf{k}}\\}$,
whereas the superscript $m$ means the Markovian approximation.

To proceed, let us note first that the generalized hydrodynamic matrix (35)
can be presented in an alternative way,

$\mathbf{T}^{(s)}(k)=\mathbf{F}^{(s)}(k)\left[\tilde{\mathbf{F}}^{(s)}(k,0)\right]^{-1},$
(36)

where the second factor denotes the inverse matrix to that constructed on the
corresponding TCFs at $z=0$. The expression (36) is more convenient, since
$\tilde{\mathbf{F}}^{(s)}(k,0)$ can be presented in terms of the SCFs and
relaxation times of the hydrodynamic origin [3, 4, 5]. Note that all the
quantities needed for calculations of matrix elements of $\mathbf{T}^{(s)}(k)$
can be obtained from computer simulations.

The solution of (34) can be presented in an analytic form via the eigenvalues
$z_{\alpha}(k)$ and eigenvectors $||\hat{X}_{i,\alpha}(k)||$ of the matrix
$\mathbf{T}^{(s)}(k)=||T_{ij}(k)||$,

$\sum\limits_{j=1}^{M}T_{ij}(k)\hat{X}_{j,\alpha}(k)=z_{\alpha}(k)\hat{X}_{i,\alpha}(k),$
(37)

where the summation runs from 1 up to the modes number $M$. It is obvious that
$M$ is equal to the number of all the dynamical variables
$\hat{\mathbf{Y}}^{(s)}$. Moreover, the eigenvalues of $\mathbf{T}^{(s)}(k)$
can be either real or complex conjugate. In the former case they define the
purely relaxing modes, whereas in the latter case they correspond to the
propagating collective excitations.

In terms of the eigenvalues/eigenvectors, the solution of (34) can be written
down as

$\tilde{F}_{ij}^{(m,s)}(k,z)=\sum\limits_{\alpha=1}^{M}\frac{G_{\alpha}^{ij}(k)}{z+z_{\alpha}(k)}.$
(38)

The numerators in (38) are the weighted amplitudes

$G_{\alpha}^{ij}(k)=\sum\limits_{l=1}^{M}\hat{X}_{i,\alpha}\hat{X}^{-1}_{\alpha,l}F^{(s)}_{lj}(k),$
(39)

which determine the contribution of each particular collective excitation
(mode $z_{\alpha}(k)$) to the TCF $F_{ij}^{(m,s)}(k,z)$. The expression (38)
can be converted to time representation,

$F_{ij}^{(m,s)}(k,t)=\sum\limits_{\alpha=1}^{M}G_{\alpha}^{ij}(k)\exp\\{-z_{\alpha}(k)t\\},$
(40)

as a sum of weighted exponents.

Some important consequences follow from all the above:

(i) Due to the linear relations between the initial set of variables
$\hat{\mathbf{P}}_{\alpha\mathbf{k}}$ and the set of their orthogonalized
counterparts $\hat{\mathbf{Y}}_{\alpha\mathbf{k}}$, the theory can be easily
converted to the form convenient for computer simulations (without projection
operators) [5].

(ii) The unitary transformation
$\hat{\mathbf{P}}_{\alpha\mathbf{k}}\rightarrow\hat{\mathbf{Y}}_{\alpha\mathbf{k}}$
does not change the collective modes spectrum as it is shown in Ref. 43. It
means that the same modes will be observed in all TCFs considered, but their
magnitudes depend on the values of the corresponding amplitudes in (40).

(iii) Within the GCM approach, one can control the results for TCFs that
follow from (40) via some sum rules for the frequency
$\langle\omega_{l}(k)\rangle=\int_{-\infty}^{\infty}d\omega\,\omega^{l}\tilde{F}^{(m,s)}(k,\omega)$
and time
$\langle\tau_{l}(k)\rangle=\int_{0}^{\infty}d\tau\,\tau^{l}F^{(m,s)}(k,\tau)$
moments of the corresponding TCFs. It has been proven in Ref. 3 that the first
($2s+1$) frequency moments along with the zeroth time moment for each of the
TCFs, calculated in the framework of the GCM approach, coincide with those
associated with the corresponding genuine functions $F_{ij}(k,t)$. Therefore,
in order to reproduce the higher frequency moments explicitly, one has to
expand the set of dynamic variables $\hat{\mathbf{P}}_{\alpha\mathbf{k}}$ by
including the higher order time derivatives.

(iv) At any level of description one can take into account the non-Markovian
effects by using the recurrent relations (33). Such a possibility was
considered in our papers [44, 45, 46, 47], using the concept of converging of
the relaxation times in the memory functions of the highest order. Formally
this is taken into account by approximation
$\tilde{\varphi}_{s-1}(z)\approx\tilde{\varphi}_{s}(z)$ in the recurrent
relation (33)). It is shown that such a modification of the GCM approach
allows to reproduce correctly one extra frequency moment even without
expanding the basic set of variables. This modification of the GCM approach is
closely related to the above mentioned spectrum of non-Markovianity [41, 42].

In the next Section various dynamic phenomena, arising in the fluids, are
analysed within the framework of GCM.

###  5 Some problems related to the fluid dynamics

####  1 Single particle diffusion

Let us start our consideration of some puzzling phenomena in the fluid
dynamics from the simplest example of TCFs: the single-particle velocity
autocorrelation function (VAF), calculated usually in MD simulations. It is
interesting that this function behaves quite differently in low- and high-
density systems, where the so-called cage effect can be observed. The cage
effect reflects a single particle trapping by the local configuration of its
neighbouring molecules, which usually manifests itself as the negative domain
(see Fig. 9) of the VAF

$\psi(t)=\frac{1}{3}\langle\mathbf{v}_{i}(t)\cdot\mathbf{v}_{i}(0)\rangle,$
(41)

where $\mathbf{v}_{i}(t)$ denotes the $i$-th particle velocity at the time
instant $t$, and $\langle\ldots\rangle$ means the thermal averaging.

Figure 9:  Velocity autocorrelation function for the 6:12 Lennard-Jones fluid
at various densities and temperatures, as obtained from computer simulation
experiment [48]. Reduced units are $n^{*}=n\sigma^{3}_{LJ}$,
$T^{*}=k_{B}T/\varepsilon_{LJ}$,
$t^{*}=t(\varepsilon_{LJ}/M\sigma_{LJ}^{2})^{1/2}$.

Since the self-diffusion coefficient $D$ and the VAF (41) are connected by a
simple relation $D=\int_{0}^{\infty}\psi(t)dt$, it means that at high density
some particles are trapped by the surrounding molecules, and its motion to the
neighbouring positions of local equilibrium is hampered to a great extent.

Within the GCM approach such a behaviour of the VAFs can be well described
[49] even at the three-variable basis
$\hat{\mathbf{P}}^{(3)}=\left\\{n(\mathbf{k},t),\dot{n}(\mathbf{k},t),\ddot{n}(\mathbf{k},t)\right\\}$,
where the dynamical variable
$n(\mathbf{k},t)=\exp(i\mathbf{k}\mathbf{r}_{j}(t))$ defines the self-
intermediate scattering function $F(k,t)=\langle
n(\mathbf{k},t)n(-\mathbf{k},0)\rangle$. The modes’ analysis shows [49] that
there are two propagating kinetic modes if $\displaystyle
mD\Omega_{E}<2k_{B}T$, where $\Omega_{E}$ denotes the so-called Einstein
frequency defined by the “force-force” SCF. Like with other kinetic modes, the
amplitudes of these single-particle excitations vanish in the hydrodynamic
limit $k\to 0$, thereby yielding the well known single exponential dependence
$F(k,t)=\exp(-Dk^{2}t)$.

Using the relation

$\psi(t)=\lim\limits_{k\to
0}\left\\{-\frac{1}{k^{2}}\frac{d^{2}}{dt^{2}}F(k,t)\right\\}$

between the VAF and self-intermediate scattering function, we can express
$\psi(t)$ as a weighted sum of two oscillating terms [49] with damping
$\tilde{\sigma}$ and renormalized frequency $\tilde{\omega}$,

$\displaystyle\tilde{\sigma}=\frac{\tau_{\mathrm{dif}}}{2\tau^{2}_{\mathrm{vib}}},\quad\tilde{\omega}=\Omega_{E}\left[1-\left(\frac{\tau_{\mathrm{dif}}}{2\tau_{\mathrm{vib}}}\right)^{2}\right]^{1/2},$
(42)
$\displaystyle\tau_{\mathrm{dif}}=\frac{mD}{k_{B}T},\quad\tau_{\mathrm{vib}}=\frac{1}{\Omega_{E}},$

where two specific times $\tau_{\mathrm{dif}}$ and $\tau_{\mathrm{vib}}$
characterize, respectively, the hydrodynamic processes, connected with the
self-diffusion, and the processes of vibrational nature, strongly dependent on
an arrangement of the tagged particle. It is obvious that the condition
$\tau_{\mathrm{dif}}\ll\tau_{\mathrm{vib}}$ presumes the oscillating behaviour
with a well-defined minimum of the VAF; conversely, if the times are close,
the cage effect cannot be observed. So, one can conclude that even in the
single-particle dynamics of fluids the collective properties in local
arrangement play an important role.

Figure 10: VAFs of the liquid Ne at the densities $\rho=$2190 kg/m3 (a), 1400
kg/m3 (b) and 691 kg/m3 (c). The solid lines correspond to the MD data. Other
lines correspond to the VAFs, calculated with the help of (33) on the 3-rd
(dotted line), 4-th (dashed-dotted line) and 5-th (dashed line) hierarchy
levels. The symbols correspond to the VAFs, obtained within Markovian
approximation for memory functions of the 3-rd (squares), 4-th (bullets) and
5-th (stars) levels. In the insets: time behavior of the corresponding VAFs at
small $t$. The results are obtained in Ref. 47.

At the quantitative level, these results can be significantly improved in
comparison with those obtained in MD simulations, by taking into account the
non-Markovian effects. For this purpose one can put
$\tilde{\varphi}_{s}(z)\approx\tilde{\varphi}_{s+1}(z)$ in the recurrent
relations (33). The obtained results (see Ref. 47) are presented in Fig. 10.
One can draw several conclusions from it. First of all, a convergence of all
results to the MD data (denoted by the solid master curve) with increasing of
the approximation order is clearly noticeable. Both lines (non-Markovian
effects) and symbols (Markovian approximation) follow the solid black line the
better, the higher order hierarchy $s$ is taken into account. It is quite
expected, since by increasing $s$ we ensure more frequency moments of the VAFs
to be satisfied.

Moreover, the modified version of the GCM approach with the effective
summation of the continued fraction, based on the assumption
$\tilde{\varphi}_{s}(z)\approx\tilde{\varphi}_{s+1}(z)$, allows one to look
from a new viewpoint at the formation of the long-time tails of VAFs $\sim
t^{-3/2}$. Recently, a microscopic interpretation of this phenomenon was
suggested from analysis of the memory kernels [50]. It was shown that the
hydrodynamic added mass, defined via the memory kernel, is negative, and the
backflow of neighbours tended to drag the particle in the direction of its
initial velocity, i.e., contributed negatively to the friction. The approach
proposed in [44, 45] and developed in [46, 47] allows to make one further step
and to evaluate the transition time to the long-time tails appearance.555The
effective time between particle collisions can be related to the instant, when
the “force–force” autocorrelation function changes its sign. Besides, the well
depth in the above mentioned function strongly decreases with decreasing
density, like it also happens in the VAFs. This phenomenon can be explained in
the following way: at low densities, the particle changes its direction of
motion due to the multiple low-angle scattering, whereas at high densities,
the “head-on” collisions dominate, yielding the profound minimum in
“force–force” autocorrelation function. This is another manifestation of a
close relationship between the single particle dynamics and the collective
behavior of its surrounding.

This issue has been elaborated in much more detail in Ref. 51, where the
transition regimes and the shapes of the VAFs are studied for the binary
mixture of particles with large mass asymmetry. The mixtures with the light
particles concentration $x=0.2$ and various ratios $\mu$ of heavy to light
particle masses were considered.

Figure 11:  Points: MD data of the MSD of the light species divided by time
for $x=0.2$ and mass ratios $\mu=2$ (red), $\mu=20$ (purple), $\mu=500$
(blue), and $\mu=\infty$ (green). Solid lines: mean field results, fitted for
the mean free path length $l$, which are 0.15, 0.15, 0.155, and 0.164,
respectively. Dashed lines: diffusion coefficients obtained from the Green-
Kubo relation multiplied by 6. The dimensionless density and temperature equal
$\varrho=0.9$ and $T=1$, respectively [51].

In Fig. 11, the mean square displacement (MSD) to time ratio is plotted as a
function of $t$. In general, one can distinguish three regimes of the MSD: the
quadratic regime for small times, where the particles move ballistically at
constant velocity, the linear regime with the usual diffusion coefficient
$D_{\alpha}=\lim\limits_{t\to\infty}\langle\Delta r_{\alpha}^{2}\rangle/6t$
for large times, and the intermediate region of anomalous diffusion. Such
diffusion is often explained by the cage effect, when particles are trapped
inside the cage formed by their surrounding neighbours for some time before
they can escape it and diffuse in the usual way. As seen from Fig. 11, the
region of the anomalous diffusion increases with $\mu$. This supports the idea
[52] that the trajectories of the light particles for large enough $\mu$
change from relatively smooth (Gaussian-like process) to intermittent ones
with a large amplitude of displacements (highly non-Fickian process or
activated hopping), as it is shown in Fig. 12.

Figure 12:  Displacement of the light particle in the mixture with
$\mu=\infty$. The other system parameters are the same as in Fig. 11[51].

It is clearly seen that the particle is repeatedly trapped, oscillating around
a certain point with an amplitude smaller than the particle size, before it
hops again to some other trap. Hence its displacement demonstrates well the
discussed above intermittent process, which is known to be a specific feature
of the self-diffusion [51].

It is also interesting to look at the time dependence of the VAFs of light and
heavy particles.

Figure 13:  Normalized VAFs of the light (left panel) and the heavy particles
(right panel) at different mass ratios $\mu$ [51].

It is seen in Fig. 13 that the VAF of the light particles shows a distinct
negative minimum, which becomes more pronounced if $\mu$ increases. On the
other hand, in the VAF of the heavy particles the minimum vanishes with
increasing mass ratio $\mu$. The position of the minimum $t_{min}$ offers a
way to estimate the typical size of a cage. With the mean thermal velocity of
$v_{0}=3$ at $T=1$ the light particles on average travel a distance
$v_{0}t_{min}=0.26$ until they are reflected by the surrounding cage. Together
with the particle radius of 0.5 this results in the cage diameter of about
1.5.

It has also to be stressed that in Ref. 51 the extensive numerical simulations
have been performed in order to verify whether the traps are indeed minima of
the potential energy landscape created by the fixed particles. Thus this study
gives in fact an example that a rather stable “solvent cage” can be formed in
the mixtures just because of a strong mass asymmetry effect.

####  2 Viscoelastic properties of fluids

The GCM framework, which manifested its efficiency for investigation of the
single particle excitations in the VAFs of fluids, can be just as effectively
applied for collective phenomena. We start from the simplest case of
transverse dynamics in simple liquids. It is well known from numerous MD
simulations that a crossover from the viscous to elastic behaviour is observed
[53, 54, 55] in the shape of the transverse spectral function that can be
treated as a signal of the shear waves appearance.

The simplest dynamic model that allows us to understand such a type of
crossover can be obtained if we consider the set of two dynamic variables, one
of which is dealt with the transverse current density
$\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}$, and the other one is related to the
first derivative of this variable
$i\hat{L}\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}$. In such a case, the equations
of macrodynamics can be written down as follows:

$\displaystyle
i\omega\langle\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}\rangle^{\omega}-\langle
i\hat{L}\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}\rangle^{\omega}=0,$ (43)
$\displaystyle\left[i\omega+\frac{1}{\tau(k)}\right]\langle
i\hat{L}\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}\rangle^{\omega}+\Gamma_{0}(k)\langle\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}\rangle^{\omega}=0,$

where the inverse relaxation time $1/\tau(k)|_{k\to 0}\simeq G/\eta$ is
related to the memory kernel, constructed on the variables
$i\hat{L}\hat{\mathbf{J}}^{\perp}_{\mathbf{k}}$, whereas
$\displaystyle\Gamma_{0}(k)=k^{2}G(k)/\varrho$ is dealt with the generalized
rigidity modulus $G(k)$, defined by the “stress-stress” SCF,

$G(k)=\displaystyle\frac{n}{k_{B}T}\left\langle\hat{\sigma}^{\perp}_{\mathbf{k}}\hat{\sigma}^{\perp}_{-\mathbf{k}}\right\rangle,$

$\varrho=nm$ and $\eta$ are the mass density and the shear viscosity,
respectively.

A similar chain of equations can be written down for the TCFs
$F_{J^{\perp}J^{\perp}}(k,t)$ built on the transverse components of the
current density, and the solution for its normalized value in the time
representation looks as follows:

$\displaystyle\frac{F_{J^{\perp}J^{\perp}}(k,t)}{F_{J^{\perp}J^{\perp}}(k)}=-\frac{z_{-}(k)}{z_{+}(k)-z_{-}(k)}e^{-z_{+}(k)t}+\frac{z_{+}(k)}{z_{+}(k)-z_{-}(k)}e^{-z_{-}(k)t},$
(44)

where the collective modes $z_{\pm}(k)$ are defined by the expression

$z_{\pm}(k)=\frac{1}{2\tau(k)}\pm\left[\frac{1}{4\tau^{2}(k)}-\frac{k^{2}}{\varrho}G(k)\right]^{1/2}.$
(45)

It can be easily verified that the first three sum rules for genuine TCF
$F_{J^{\perp}J^{\perp}}(k,t)$ are satisfied by the expression (45). Moreover,
the hydrodynamic correlation time for $F_{J^{\perp}J^{\perp}}(k,t)$ for this
simplest non-trivial model (43) can be calculated within the GCM approach.

In the hydrodynamic limit $k\to 0$, the first term in Eq. (44), being
proportional to $k^{2}$, tends to zero, and the “current-current” TCF is
reduced to its well-known single exponent form of a purely diffusive nature,

$\frac{F_{J^{\perp}J^{\perp}}(k,t)}{F_{J^{\perp}J^{\perp}}(k)}\simeq\exp\left(-\frac{\eta}{\varrho}k^{2}t\right).$
(46)

Contrary, at $k>k_{H}=\sqrt{\varrho G}/2\eta$ there exist two propagating
modes, as it is clearly seen from (45). Therefore, the kinetic propagating
modes related to the shear waves are detectable [53, 54] in the transverse
“current-current” TFC, starting from some threshold wavenumber
$k>\sqrt{2}k_{H}$.

Figure 14:  The Fourier-transforms of transverse “current-current” TCFs of
liquid Cs at $\varrho=1832$ kg/m3 and $T=308$ K for six values of $k$. The
time and length scale units are $\tau_{\sigma}=4.494$ ps and
$\sigma=k_{min}^{-1}=6.239\AA$, respectively.

The above mentioned phenomenon is illustrated in Fig. 14, where the Fourier
spectra of the transverse “current-current” TCF of liquid Cs, obtained in Ref.
54 within the GCM approach for the dynamic model with four variables, $M=4$,
are shown. For $k<\sqrt{2}k_{H}$, the functions $C^{t}(k,\omega)$ are single
Lorentzians (cf. Eq. (46)) centred at the origin.666At higher modes
approximation, the value $k_{H}$ cannot be obtained analytically, and only
numerical calculation of the $T(k)$ matrix spectrum is possible. For
$k>\sqrt{2}k_{H}$, the functions $C^{t}(k,\omega)$ have a peak centred at
$\omega>0$. The strong contribution of the pronounced non-central Lorentzian
is the consequence of propagating modes with small damping associated with the
real part of the relevant eigenvalue $z(k)$. As the damping coeffcient
increases for $k>1$ Å-1, the peak structure of $C^{t}(k,\omega)$ becomes less
pronounced, yielding the corresponding reduction of the well depth in the TCF
$F_{J^{\perp}J^{\perp}}(k,t)$. Though the existence of a negative domain of
the transverse “current-current” TCF resembles that in VAFs of the fluids at
large densities (see Sec. 5.1), its meaning is obviously different: whereas
the VAFs minima describe an essentially single particle cage effect, the
minima of TCF $F_{J^{\perp}J^{\perp}}(k,t)$ correspond to the collective shear
waves. However, the physical origin of such specific behaviour in the both
cases could be similar and is connected with the phenomenon of local ordering
in the fluid. At the same time, the characteristic length of the locally
ordered domain can be estimated as $1/k_{H}$.

Figure 15: The spectrum of generalized collective modes: imaginary (left
panel) and real (right panel) parts of reduced eigenvalues at $M=4$. Dotted
curve displays the hydrodynamic behaviour $z^{h}$.

It is also instructive to look at the collective excitation spectrum,
presented in Fig. 15. For $k>0.092$ Å-1 the spectrum of transverse excitations
consists of two pairs of complex conjugated propagating modes. The high-
frequency and low-frequency modes have comparable real parts (damping) of the
eigenvalues only at $k>1$ Å-1. For smaller $k$, the contribution from the
upper mode into all dynamical processes is very small, and in the hydrodynamic
limit it almost vanishes. At $k<0.092$ Å-1 the lower-lying propagating mode
disappears and transforms into two relaxing modes with purely real
eigenvalues. In the $k\to 0$ limit, the eigenvalue of one of these modes tends
to a finite damping coefficient, while the second eigenvalue behaves as
$\mbox{Re}\,z^{h}=\displaystyle\eta k^{2}/\varrho$ in a full agreement with
the hydrodynamic theory. It should be also emphasized that in highly viscous
fluids as well as in glass-like systems [56] with large shear viscosity
coefficient the range of hydrodynamic behaviour is very small ($k_{H}\sim
1/\eta$), so that such fluids behave like elastic bodies up to the
characteristic length $\sim 1/k_{H}$.

One has to mention that from the mathematical viewpoint the model (43) is
rather general and can be used to study many other cases related to the
dynamics of conserved quantities. For instance, within a similar two-variable
model the formation of thermal waves in fluids can be explained taking into
account the coupling between the heat density and the density of heat current
[29, 57]. That indicates a certain universality of the above described
scenario at the heat waves formation, optic-like modes in mixtures (see Sec.
5.3), spin waves [58] in magnetic liquids, etc.

Before we start considering other kinds of propagating modes, let us conclude
Sec. 5.2 by one more demonstration of the viscoelastic behaviour in fluids.

Figure 16:  Normalized “stress-stress” autocorrelation function $\eta_{2}(t)$
for the heavy component of the binary system with concentration $x=0.2$ of the
light species, dimensionless density $\varrho=0.6$ and temperature $T=1.05$,
respectively [51].

In Fig. 16 we plot the “stress-stress” autocorrelation function of the heavier
component (labelled here by 2)

$\eta_{2}(t)=\left\langle\sigma_{xy}^{(2)}(t)\sigma_{xy}^{(2)}(0)\right\rangle$
(47)

of the binary Lennard-Jones mixture studied in Ref. 51.

It is obvious that with increasing mass ratio $\mu$, the relaxation times grow
very similarly to the VACF of the heavy component $\psi_{2}(t)$ (see the right
panel of Fig. 13). However, unlike $\psi_{2}(t)$, the “stress-stress”
autocorrelation function exhibits a distinct plateau, which can be considered
as a forerunner of the glass phase formation. Such a delayed relaxation of
$\eta_{2}(t)$ anticipates the large values of the fluid viscosity, which is
expressed by the Green-Kubo relation from the corresponding “stress-stress”
autocorrelation function. So one can conclude that the strongly asymmetric
binary mixture behaves in a very glassy-like manner, as far as its viscosity
is concerned. Obviously, more complicated dynamic models are needed to
describe such behaviour.

The last issue, which we consider in Sec. 5.2, is a possibility to observe the
non-Newtonian behaviour in simple fluids. Using the two-variable model (43)
developed within the linear response theory [8, 26], one can relate the mean
value of the transverse stress tensor $\langle\sigma_{T}\rangle^{\omega}$ to
the strain $\gamma_{T}(\omega)$ as follows:

$\langle\sigma_{T}\rangle^{\omega}=i\omega\eta(\omega)\gamma_{T}(\omega),$
(48)

where $\eta(\omega)$ is the generalized shear viscosity that has the Lorenz-
like form

$\eta(\omega)=\frac{\eta}{i\omega\tau+1}$ (49)

with the corresponding relaxation time $\tau=\eta/G$. The expressions
(48)-(49) allow one to write down

$\displaystyle\langle\sigma_{T}\rangle^{t}\simeq\eta\frac{\partial\gamma_{T}(t)}{\partial
t}$ for times $\displaystyle\quad t\gg\tau,$ (50)
$\displaystyle\langle\sigma_{T}\rangle^{t}\simeq G\gamma_{T}(t)$ for times
$\displaystyle\quad t\ll\tau.$

Therefore, at large times the fluid behaves as a Newtonian liquid (the stress
is defined by the strain rate), whereas at small times the Hooke’s law is
obeyed (see the second equation in (50)), and the fluid dynamics resembles
that of the elastic body.

####  3 Optical phonon-like excitations and fast sound

Let us consider now some problems typical for the collective dynamics of fluid
mixtures. The most interesting questions that arise in this case are the
following: is there really a strong resemblance between the collective
excitations in fluids and the phonon dynamics in crystals? Could the so-called
“optic-like collective excitations” be observed experimentally?

It is well known that the optic-like phonon excitations in solids describe
opposite motions of particles in different species. So, if the cage effects
are well pronounced in the fluid, and the nearest neighbours of a particle are
formed mainly by the particles of different species, one may hope to find such
collective modes in experiments. Otherwise, from the standard hydrodynamics we
know that in the hydrodynamic limit only one pair of propagating modes exists
in a fluid mixture, and these modes are associated with acoustic sound
excitations describing (like in crystals) the coherent movements of all
particles. How should the theory be modified in order to treat this problem in
more details?

We start our treatment from the case of binary mixtures. For this purpose we
introduce a new dynamic variable defined as a normal coordinate to the density
of total current
$\hat{\mathbf{J}}_{t,\mathbf{k}}=\hat{\mathbf{J}}_{\mathbf{1},k}+\hat{\mathbf{J}}_{2,\mathbf{k}}$,
defined as a sum of its partial components. A new dynamic variable can be
introduced via the mass-concentration current according to

$\hat{\mathbf{J}}_{x,\mathbf{k}}=x_{2}\hat{\mathbf{J}}_{\mathbf{1},k}-x_{1}\hat{\mathbf{J}}_{\mathbf{2},k}.$
(51)

Note that $x_{a}=m_{a}N_{a}/\sum_{a}m_{a}N_{a}$ with $a=1,2$, and the new
variable $\hat{\mathbf{J}}_{x,\mathbf{k}}$ is orthogonal to
$\hat{\mathbf{J}}_{t,\mathbf{k}}$ in the sense that the corresponding SCF is
equal to zero,
$\left(\hat{\mathbf{J}}_{t,\mathbf{k}},\hat{\mathbf{J}}_{x,\mathbf{k}}\right)=0$.
This means that these dynamic variables are weakly coupled at the equilibrium,
and some simplified dynamic models can be used for the subsequent analysis.

Thus using the two-variable model
$\hat{\mathbf{P}}_{x}^{\perp}=\left\\{\hat{J}_{x,\mathbf{k}}^{\perp},\dot{\hat{J}}_{x,\mathbf{k}}^{\perp}\right\\}$,
it is straightforward to find the condition (see Sec. 5.2), when the
transverse propagating mass-concentration waves appear. In the hydrodynamic
limit this condition looks as follows [59]:

$\delta_{x}^{\perp}=\frac{\bar{\omega}^{\perp}_{2,J}(0)D_{12}^{2}S_{xx}^{2}(0)\bar{m}^{2}}{4(x_{1}x_{2}k_{B}T)^{2}}<1,$
(52)

where $\bar{\omega}^{\perp}_{2,J}(0)=\lim\limits_{k\to
0}\left\langle\dot{\hat{J}}_{x,\mathbf{k}}^{\perp}\dot{\hat{J}}_{x,-\mathbf{k}}^{\perp}\right\rangle/\left\langle\hat{J}_{x,\mathbf{k}}^{\perp}\hat{J}_{x,-\mathbf{k}}^{\perp}\right\rangle$
denotes the normalized second-order frequency moment; $D_{12}$ means the
mutual diffusion coefficients; $S_{xx}(0)$ denotes the mass-concentration
static structure factor at $k=0$, and $\bar{m}=\sum_{a}m_{a}c_{a}$,
$c_{a}=N_{a}/N$.

In the long-wavelength limit a similar condition has been derived [60] for the
longitudinal dynamics of binary mixtures within the three-variable model with
the set of dynamic variables
$\hat{\mathbf{P}}_{x}^{||}=\left\\{\hat{n}_{x,\mathbf{k}},\hat{J}_{x,\mathbf{k}}^{||},\dot{\hat{J}}_{x,\mathbf{k}}^{||}\right\\}$.
In this case, in addition to the pair of the propagating excitations, the
appearance of which is described by a condition like (52), there exists a
purely diffusive mode related to the mutual diffusion process. In the
generalized version, when the $k$-dependence is taken into account, this
condition can be written down in the form

$\delta_{x}^{||}(k)=\frac{\bar{\omega}^{||}_{4,x}(k)}{[2\tau_{xx}(k)\bar{\omega}^{||}_{2,x}(k)]^{2}}<1,$
(53)

where

$\displaystyle\bar{\omega}^{||}_{2,x}(k)=\frac{k^{2}\left\langle{\hat{J}}_{x,\mathbf{k}}^{||}{\hat{J}}_{x,-\mathbf{k}}^{||}\right\rangle}{\bar{m}\left\langle\hat{n}_{x,\mathbf{k}}^{||}\hat{n}_{x,-\mathbf{k}}^{||}\right\rangle},\quad\bar{\omega}^{||}_{4,x}(k)=\frac{\left\langle\dot{\hat{J}}_{x,\mathbf{k}}^{||}\dot{\hat{J}}_{x,-\mathbf{k}}^{||}\right\rangle}{\left\langle\hat{J}_{x,\mathbf{k}}^{||}\hat{J}_{x,-\mathbf{k}}^{||}\right\rangle},$
$\displaystyle\tau_{xx}(k)=S_{xx}^{-1}(k)\int\limits_{0}^{\infty}F_{xx}(k,t),$

denote the normalized second and fourth order frequency moments respectively,
while $\tau_{xx}(k)$ is the corresponding relaxation time, determined by the
mass-concentration TCF $F_{xx}(k,t)$.

Figure 17:  $\delta^{||}_{x}(k)$ as a function of the wavenumber for KrAr and
He0.65Ne0.35. optic-like excitations exist in the long-wavelength region only
when $\delta_{x}^{||}(k)<1$. The results are obtained in Ref. 60.

The condition (53) has been verified for several binary mixtures, and some
results, obtained [60], in particular, for KrAr and He0.65Ne0.35 are shown in
Fig. 17. It is seen that in the hydrodynamic limit this condition is fulfilled
for the first mixture, but not for the second one, so that an additional pair
of propagating modes, describing the collective opposite motion of particles
in different species, could be expected in the KrAr mixture for small $k$.

Figure 18:  Dispersions of the sound ($z_{1}(k)$) and high-frequency
($z_{2}(k)$) excitations in the equimolar Lennard-Jones mixture KrAr at 116 K
and the dense gas mixture He0.65Ne0.35 at 39.3 K, calculated numerically
within the 14-variable GCM approach. Dashed and solid curves correspond to
dynamical eigenmodes obtained by separate treatment of the dynamics using the
5-variable sets with labels $x$ and $t$, correspondingly. Dotted vertical
lines approximately separate the short-wavelength regions of “partial”
behaviour of branches. The branch $z_{2}(k)$ corresponds in the long-
wavelength region to optic-like excitations in KrAr, while for He0.65Ne0.35 at
39.3 K the optic-like excitations are suppressed[60].

Reliability of the GCM approach for the analysis of mass–concentration
fluctuations is illustrated in Fig. 18, where the imaginary parts of the
corresponding propagating eigenvalues for the equimolar mixture KrAr and the
dense gas mixture with disparate masses He0.65Ne0.35 are plotted. The spectra
of the longitudinal collective excitations were obtained within the numerical
parameter-free 14-variable GCM approach [60], which included: i) total
(labelled $t$) and mass-concentration (labelled $x$) number densities, ii) the
corresponding current densities, and iii) total energy density along with the
derivatives of variables ii) and iii) up to the third order. All the elements
of the 14$\times$14 generalized hydrodynamic matrix $T(k)$ were evaluated
directly in MD simulations, so no free parameter was invoked. The number of
dynamical variables was chosen to be in agreement with the nine-variable basis
set, used for the case of simple liquids [3] with the time derivatives of the
hydrodynamic variables up to the third order, which allowed one to obtain a
very accurate description of the LJ fluid.

Let us analyse the results for the mode dynamics of both systems. The branch
$z_{1}(k)$ corresponds to the hydrodynamic sound excitations with the linear
dispersion in the small wavenumber region. The branch $z_{2}(k)$ in the long-
wavelength limit is caused by mass–concentration fluctuations and can be
reproduced by treatment of solely dynamical variables describing the
mass–concentration fluctuations.

The vertical dotted lines in Fig. 18 separate two regions of wavenumbers, in
which either the collective (small $k$) or partial (large $k$) forms of
dynamics prevail (see also Fig. 19 and the subsequent discussion). One can see
in Fig. 18 that in the case of the dense gas mixture He0.65Ne0.35 the short-
wavelength region of “partial” dynamics begins at much smaller wavenumbers
than for KrAr. Moreover, in the long-wavelength limit, in contrast to the
branch patterns in KrAr, there exists only one branch of propagating sound
excitations $z_{1}(k)$ with the linear dispersion, while the branch $z_{2}(k)$
is suppressed at $k\to 0$. The dashed and solid curves in Fig. 18 indicate
that the collective dynamics of the KrAr mixture in the short-wavelength
domain can be well described by the five-variable sets, built on the variables
i) and ii), while the influence of the energy fluctuations is not essential.
At the same time, this is not the case for the He0.65Ne0.35 system, since the
region with the “collective” dynamics is too narrow to draw any conclusion
about the mode behaviour within the five-variable models.

To analyse the collective vs. partial behaviour in the binary mixtures, it is
useful to go back to the transverse dynamics. In the general case within the
GCM approach the equation for the TCF $F_{ij}(k,t)$ can be written as

$\frac{F_{ij}(k,t)}{F_{jj}(k)}=\sum\limits_{r=1}^{n_{r}}\bar{A}_{ij}^{r}e^{-\sigma_{r}t}+\sum\limits_{p=1}^{n_{p}}\left[\bar{B}_{ij}^{p}\cos(\omega_{p}t)+\bar{C}_{ij}^{p}\sin(\omega_{p}t)\right]e^{-\sigma_{p}t},$
(54)

where the first summation runs over all of the relaxing modes $n_{r}$, while
the second one is performed over all the propagating modes $n_{p}$;
$\bar{A}_{ij}^{r}(k)$, $\bar{B}_{ij}^{p}(k)$, and $\bar{C}_{ij}^{p}(k)$ denote
the corresponding weighting coefficients; $\omega_{p}(k)$ means the dispersion
of the propagating mode, while $\sigma_{r}(k)$, $\sigma_{p}(k)$ are the
damping coefficients of the relaxing and propagating excitations,
respectively. This result can be obtained within the GCM approach for the
dynamic model with the set of $n_{r}+2n_{p}$ dynamic variables.

Figure 19: The normalized weighting coefficients $\bar{B}_{ij}^{p}(k)$ of
Mg70Zn30 for the two lowest propagating excitations $z_{1}(k)$ (closed boxes)
and $z_{2}(k)$ (open boxes) for four different TCFs. Solid and dashed lines in
the upper frame correspond to the contributions to the autocorrelation
functions $F_{tt}(k,t)$ and $F_{\mbox{\tiny{ZnZn}}}(k,t)$, respectively. In
the lower frame the solid and dashed lines correspond to the functions
$F_{xx}(k,t)$ and $F_{\mbox{\tiny{MgMg}}}(k,t)$, respectively[59].

The interplay between the “partial” and “collective” behaviour of TCFs is well
seen in Fig. 19 on the example of the eight-mode transverse dynamics in a
glass-forming molten alloy Mg70Zn30 at $T$ = 833 K, $n$ = 0.0435 Å-3. The
study has been performed [56, 59] within the GCM approach for the eight-
variable model based on the partial densities of transverse momentum and its
time derivatives up to the 3-rd order. For the wavenumbers $k>0.05$ Å-1 we
found four branches of propagating modes, i.e. $n_{r}$ = 0 and $n_{p}=4$. The
obtained results [59] allowed us to calculate the weighting coefficients (see
(54)) for several transverse TCFs being interest of. The $k$-dependences of
normalized amplitudes $\bar{B}_{ii}^{p}(k)$ with
$i=\\{t,x,\mbox{Mg},\mbox{Zn}\\}$ for the lowest two propagating modes (that
make the main contributions and are related to shear-waves and optic-like
excitations) are plotted in Fig. 19. As one can see, for large $k$ the partial
TCFs $F_{\mbox{\tiny{ZnZn}}}(k,t)$ and $F_{\mbox{\tiny{MgMg}}}(k,t)$ are
almost completely determined by the branches $z_{1}(k)$ and $z_{2}(k)$,
respectively. The same can be said about the contributions of these modes to
the TCFs $F_{tt}(k,t)$ and $F_{xx}(k,t)$ at small $k$. Thus at small $k$ the
collective transverse dynamics dominates, whereas the domain of large
wavenumbers favours the “partial” dynamics. The crossover region is well
observed nearby $k\simeq 0.7$ Å-1.

Figure 20: The spectral functions $F_{tt}(k,\omega)$ (open boxes),
$F_{xx}(k,\omega)$ (crosses), $F_{\mbox{\tiny MgMg}}(k,\omega)$ (solid line),
and $F_{\mbox{\tiny ZnZn}}(k,\omega)$ (dashed–dotted line) for Mg70Zn30 at two
different values of $k$ [59].

In Fig. 20 the transverse spectral functions $F_{tt}(k,\omega)$,
$F_{xx}(k,\omega)$, $F_{\mbox{\tiny MgMg}}(k,\omega)$, and $F_{\mbox{\tiny
ZnZn}}(k,\omega)$ for Mg70Zn30, obtained as numerical Fourier transforms of
the relevant MD-derived TCFs, are shown for two values of $k$ that belong to
the regions with dominant collective and partial types of collective
behaviour, respectively. One can see that the spectral functions of the
dynamical variables $J_{t,\mathbf{k}}^{\perp}$ and $J_{x,\mathbf{k}}^{\perp}$
for $k=$0.232 Å-1 have the one-peak structure with well-defined maxima,
positions of which correspond closely to the frequencies of collective
excitations obtained by the GCM method. The partial spectral functions for the
smaller value of $k$ have the first peak located nearly at the frequency of
the shear wave branch, which is much more pronounced in both partial spectral
functions than the shoulder (or a heavily smeared peak) associated with the
optic-like excitations with larger damping coefficient. For the larger $k$ =
2.022 Å-1, the situation is quite the opposite: the partial spectral functions
manifest a one-peak structure, while the functions $F_{tt}(k,\omega)$ and
$F_{xx}(k,\omega)$ each exhibit a main peak (nearly at the position of the
maximum for $F_{\mbox{\tiny ZnZn}}(k,\omega)$ and a shoulder (close to the
position of the maximum for $F_{\mbox{\tiny MgMg}}(k,\omega)$). This is in a
complete agreement with our discussion of the mode contributions and the
results presented in Fig. 19.

The ideas, verified for the case of binary fluids, can be developed and
applied for the study of propagating optic-like modes in a many component
mixture. In the case of a ternary mixture the set of the orthogonalized
dynamic variables can be defined in the form [61]

$\displaystyle\hat{J}_{t,\mathbf{k}}^{\perp}=\hat{J}_{A,\mathbf{k}}^{\perp}+\hat{J}_{B,\mathbf{k}}^{\perp}+\hat{J}_{C,\mathbf{k}}^{\perp},$
$\displaystyle\hat{J}_{x1,\mathbf{k}}^{\perp}=x_{A}\hat{J}_{B,\mathbf{k}}^{\perp}-x_{B}\hat{J}_{A,\mathbf{k}}^{\perp},$
(55)
$\displaystyle\hat{J}_{x2,\mathbf{k}}^{\perp}=(x_{A}+x_{B})\hat{J}_{C,\mathbf{k}}^{\perp}-x_{C}(\hat{J}_{A,\mathbf{k}}^{\perp}+\hat{J}_{B,\mathbf{k}}^{\perp}),$

where $x_{\alpha}$ with $\alpha=A,B,C$ are the corresponding mass-
concentrations. In order to analyse a crossover from the collective to partial
dynamics, the ternary Lennard-Jones mixture with the molar ratio 2:1:1 and
mass ratio 13.91:8.63:4.63 at temperature $T=116$ K and density $n$=0.0182 Å-3
has been studied within the GCM approach [61].

Figure 21: Imaginary parts of the transverse complex-conjugated eigenvalues,
obtained for the six-variable basis set $\hat{\mathbf{P}}^{(6\perp)}$
(symbols) and calculated on the separated subsets of dynamical variables
(lines) $\hat{\mathbf{P}}^{(2\perp)}_{i}$ and
$\hat{\mathbf{P}}^{(2\perp)}_{\alpha}$ for small and intermediate $k$,
respectively[61].

The collective mode spectra were calculated for the six-variable basic set
$\hat{\mathbf{P}}^{(6\perp)}=\\{\hat{J}_{A,\mathbf{k}}^{\perp},\hat{J}_{B,\mathbf{k}}^{\perp},\hat{J}_{C,\mathbf{k}}^{\perp},\dot{\hat{J}}_{A,\mathbf{k}}^{\perp},\dot{\hat{J}}_{B,\mathbf{k}}^{\perp},\dot{\hat{J}}_{C,\mathbf{k}}^{\perp}\\}$
as well as several separated subsets of dynamical variables, namely for
$\hat{\mathbf{P}}^{(2\perp)}_{\alpha}=\\{\hat{J}_{\alpha,\mathbf{k}}^{\perp},\dot{\hat{J}}_{\alpha,\mathbf{k}}^{\perp}\\}$
with $\alpha=\\{A,B,C\\}$ and
$\hat{\mathbf{P}}^{(2\perp)}_{i}=\\{\hat{J}_{i,\mathbf{k}}^{\perp},\dot{\hat{J}}_{i,\mathbf{k}}^{\perp}\\}$
with $i=\\{t,x1,x2\\}$. The obtained results for the imaginary parts of the
transverse complex-conjugated eigenvalues are shown in Fig. 21. The crossover
range nearby $k\simeq$0.8 Å-1 is well seen, such that: i) for larger $k$ the
partial dynamics dominates and the separated partial subsets
$\hat{\mathbf{P}}^{(2\perp)}_{\alpha}$ match perfectly the results for the
six-variable dynamical model; ii) at smaller wavenumbers the results for
imaginary parts of the transverse complex conjugate eigenvalues are well
reproduced with the help of the two-variable subsets
$\hat{\mathbf{P}}^{(2\perp)}_{\alpha}$ (see (3)), describing the collective
behaviour. Thus we conclude that the crossover from the collective behaviour
to partial one in fluid mixtures can be observed not only in the
$k$-dependences of weighting coefficients (see Fig. 19), but also in the
$k$-dependences of imaginary parts of propagating collective modes.

Let us return briefly to the case of binary fluids with a significant mass
difference of particles. Two important consequences of this difference are
usually observed: i) the crossover line to the partial behaviour is shifted to
smaller wavenumbers and ii) the propagating modes are strongly separated at
larger $k$. It is well seen in Fig. 18 for the He0.65Ne0.35 mixture with the
mass ratio 5.04.

Figure 22: The imaginary and real parts of the longitudinal complex
eigenvalues $z_{j}(k)$ corresponding to the frequency and damping of the
propagating collective excitations, respectively [62].

In Fig. 22 the results, obtained in Ref. 62 for the generalized propagating
collective modes of the Li4Pb molten alloy with mass ratio 29.85 at the
temperature of 1085 K and mass-density of 3556.76 kg/m3 are presented. As can
be seen from the top part of Fig. 22, in the long-wavelength region, the
frequency of the concentration propagating mode decreases sharply to the
values typical to sound excitations, but its attenuation coefficient
increases. For larger wavenumber $k$, the dispersion curves of the generalized
concentration and sound modes are strongly separated, and the regime of
partial behaviour is observed. The closeness of the frequencies of the
propagating modes in the region of small $k$ with the subsequent rapid growth
of the difference between them creates good preconditions to explore these
excitations in binary mixtures with large mass ratio in light scattering
experiments. One can trace this issue back to the 1980ies, when the first
investigations of the phenomenon, named later as the fast sound, were
performed [34] for the Li4Pb molten alloy and compared to the prediction of
the Mori-Zwanzig formalism. A new propagating mode was found to appear at
$k\sim 0.1$ Å-1, showing the linear dispersion in the wavenumber region 0.1
Å${}^{-1}\leq k\leq$ 0.6 Å-1 and having the velocity, which is more than three
times higher than that of the ordinary sound. The results of our studies (see
also [63, 62, 66]) support the conclusion of Ref. 34 about the possibility to
observe a higher-frequency propagating mode in a binary system of two species
with a large mass difference. Moreover, we establish that such propagating
excitations are in fact the propagating mass-concentration modes, which behave
in a rather specific way in a binary fluid with the large mass ratio. We note
that in most cases studied for the binary fluids with large mass ratio, it has
been found that the frequency of propagating mass-concentration mode tends to
zero at some fixed and very small $k$ (like it was observed for the frequency
of the generalized transverse modes discussed in Sec. 5.2).

For another large group of binary mixtures, it was found that the frequency of
propagating mass-concentration modes in the long-wavelength limit tends to a
finite non-zero value, so one has a reason to talk about the optic-like
excitations. The question about the possibility of its experimental
observation is discussed in more detail for the case of molten salts.

####  4 Dynamics of molten salts: optic-like excitations

From the theoretical point of view the main difference in the description of
molten salts in comparison with binary mixtures of neutral particles is the
nature of the inter-particle interactions. Since a molten salt can be
considered as a system of charged particles or ions, the interactions between
them are described by the long-range Coulomb potential. Besides, in a molten
salt the electroneutrality condition $\sum_{\alpha}q_{\alpha}n_{\alpha}=0$
should be satisfied, and this imposes certain restrictions on the
concentrations of ions of different charges. In this case, it is more
convenient to use a dynamic variable that is associated with the charge
density $\hat{\mathbf{J}}_{q,\mathbf{k}}$ instead of its mass-concentration
counterpart (51), which is usually used for the binary mixtures of neutral
particles. Because of the electroneutrality condition one has

$\hat{\mathbf{J}}_{q,\mathbf{k}}=q_{1}\hat{\mathbf{J}}_{\mathbf{1},k}+q_{2}\hat{\mathbf{J}}_{\mathbf{2},k}=\Bigg{[}\frac{q_{1}}{m_{1}}-\frac{q_{2}}{m_{2}}\Bigg{]}\hat{\mathbf{J}}_{x,\mathbf{k}}.$
(56)

Thus one can use our previous results obtained for the binary mixture of
neutral particles, in particular, the conditions (52) and (53). Due to the
long-range character of Coulomb interaction, one has the condition
$S_{qq}(k)\sim k^{2}$ for the “charge-charge” SCF in the hydrodynamic limit.
This means that the conditions (52) and (53) are always fulfilled at small
wavenumbers $k$. Therefore, the optic-like propagating modes always exist in
molten salts.

In order to illustrate this statement and to discuss some specific features of
the collective excitations in molten salts, the results obtained for the
liquid LiF system in Ref. 30 are displayed in Fig. 23.

Figure 23:  The dispersion of propagating collective excitations in molten LiF
at 1287 K obtained in the eight-variable model (spline-interpolated solid
curves). Symbols are the results obtained for separate sets of three dynamical
variables. The dashed line shows the linear dispersion for sound excitations
with $c=5320$ m/s[30].

The collective modes analysis was performed for the eight-variable basic set,
including the mass and charge densities, mass and charge current densities,
energy densities as well as derivatives of the last three dynamical variables.
Besides, the four separate sets
$\hat{\mathbf{P}}^{(3i)}_{\mathbf{k}}=\left\\{\hat{n}_{i,\mathbf{k}},\hat{J}^{||}_{i,\mathbf{k}},\dot{\hat{J}}^{||}_{i,\mathbf{k}}\right\\}$,
$i=\\{m,q,\mbox{Li},\mbox{F}\\}$ (the labels $t$ and $q$ correspond to mass
and charge, respectively), were also taken into consideration to analyse the
collective modes spectra in more details.

In Fig. 23 the imaginary parts of two complex eigenvalues $z_{\alpha}(k)$,
which correspond to the propagating excitations, are shown. Another branch of
propagating excitations, corresponding to heat waves, was obtained in the
region $k>0.8$ Å-1; however, here we focus on the propagating density
fluctuations only. The dispersion curves shown by solid spline-interpolated
lines were estimated from the eight-variable model. One can conclude that in
the region $k<1.4$ Å-1 the high-frequency branch is solely defined by
propagating charge waves, while the low-frequency branch comes from the
fluctuations of the total mass density and in the long-wavelength region shows
an almost linear dispersion law with the propagation speed $c=5320$ m/s. In
this region the low- and high-frequency branches correspond to acoustic and
optical phonon-like excitations, respectively. For larger wavenumbers $k>1.4$
Å-1, the partial behaviour of both branches was established, with the low and
high-frequency branches describing solely the heavy (F) and light (Li)
subsystems in the melt. Like in the case of non-ionic mixtures (cf. Fig. 21),
the separate three-variable sets, labelled by $t$ and $q$, perfectly fit the
data of the eight-variable model at small $k$, whereas at large wavenumbers
the results obtained using the reduced basic sets, labelled by F and Li, show
a good agreement with those of the extended eight-variable model. Another
difference from the non-ionic mixtures is that the collective excitations in
the molten LiF salt are well separated in frequency at all $k$, while for the
mixture of neutral particles it is not the case (see Fig. 18 for comparison).

Figure 24:  The charge density autocorrelation function for molten LiF
obtained in MD simulations (solid curve) and its GCM replicas estimated from
the eight-variable (dashed curve) and three-variable (dotted curve)
models[30].

In Fig. 24 we show two GCM replicas for the smallest wavenumber charge density
autocorrelation function, obtained with the extended eight-variable set of
dynamic variables (dashed curve) and the reduced three-variable one (dotted
curve), no fitting parameters in both cases. The GCM replica obtained from the
three-variable treatment of the charge subsystem reproduces the oscillating
behaviour of $F_{qq}(k,t)$, but contains oscillations that are less
overdamped, because the interaction between charge fluctuations and other
hydrodynamic processes was not taken into account. In the case of the eight-
variable treatment, the GCM data fit the MD results at small and intermediate
times well enough.

As it has been already mentioned, one of the advantages of the GCM approach is
the possibility to separate the mode contributions to the GCM replica, see,
for instance, Eq. (54).

Figure 25:  Mode amplitudes of different relaxing and propagating modes for
the charge density autocorrelation functions obtained for molten LiF in the
eight-variable GCM model (symbols connected by lines)[30].

In Fig. 25 the mode amplitudes for the two main relaxation processes of the
mutual diffusion $A^{q}_{qq}(k)$, the thermal diffusivity $A^{th}_{qq}(k)$ as
well as the two symmetric contributions coming from optical and acoustic
branches of collective excitations are shown by symbol-connected lines for the
case of eight-variable model of collective dynamics in molten LiF. It is seen
that for $k<0.5$ Å-1 neither the sound excitations nor the thermal diffusivity
contribute to the “charge–charge” TCF $F_{qq}(k,t)$, and the shape of
$F_{qq}(k,t)$ is determined solely by the relaxation process of the electric
conductivity and propagating charge waves. Moreover, the contribution from the
non-hydrodynamic charge waves is almost four times as large as that from the
hydrodynamic relaxation process. This shows the striking difference between
the molten salts with the long-range interaction and non-ionic liquid
mixtures, for which the mode strength of optic-like excitations in
$F_{xx}(k,t)$ vanishes rapidly towards small wavenumbers [60].

With this in mind, let us try to answer the question: is it possible to
observe the optic-like excitations in scattering experiments considering the
shape of dynamic structure factor? In Ref. 66 the dynamic properties and the
crossover between hydrodynamic and kinetic modes in liquid alloys Na57K43 were
investigated by applying the GCM analysis scheme and using the inelastic X-ray
scattering (IXS) data. The GCM scheme was built using the extended eight-
variable formalism with
$\hat{\mathbf{P}}^{(8)}=\\{\hat{n}_{t,\mathbf{k}},\hat{n}_{x,\mathbf{k}},\hat{J}_{t,\mathbf{k}}^{||},\hat{J}_{x,\mathbf{k}}^{||},\hat{\varepsilon}_{\mathbf{k}},\dot{\hat{J}}_{t,\mathbf{k}}^{||},\dot{\hat{J}}_{x,\mathbf{k}}^{||},\dot{\hat{\varepsilon}}_{\mathbf{k}}\\}$
as well as the reduced three-variable subsets
$\hat{\mathbf{P}}^{(3\alpha)}=\\{\hat{n}_{\alpha,\mathbf{k}},\hat{J}_{\alpha,\mathbf{k}}^{||},\dot{\hat{J}}_{\alpha,\mathbf{k}}^{||}\\}$,
$\alpha=\\{t,x,\mbox{Na},\mbox{K}\\}$. Usage of the different sets of dynamic
variables allows us to ascertain the origin of each branch of collective
excitations in the spectrum and processes responsible for their appearance in
various wavenumber domains.

Figure 26: Experimental dispersion relation [66] obtained from the maxima of
$C_{\mbox{\tiny{IXS}}}(Q,\omega)$ ($\times$), along with the hydrodynamic
dispersion derived by ultrasound measurements [67] (black dashed line). The
two propagating modes predicted by complete eight-variable GCM theory (high
frequency, $\circ$; low frequency, $\bullet$), and the outcome of partial
3-variable GCM analysis with total (blue dashed line), concentration (pink
dashed line), Na (red solid line), and K (green solid line) variables are also
reported.

The result for spectral current density is reported in Fig. 26 and clearly
shows that the partial type of dynamics (full red and green lines for the Na
and K subsets, respectively) is dominant for $Q>5$ nm-1. The corresponding
excitations are very close to the two eigenvalues of the eight-variable
treatment in this range, while at smaller $Q$ the same eigenvalues are
reproduced by the total density and concentration subsets. This is a clear
indication of the existence of two dynamical regimes: the collective region at
low $Q$ and the domain of partial dynamics at higher $Q$, in a complete
agreement with our previous conclusions (see also [68]).

Figure 27: Classical resolution deconvoluted $S(Q,\omega)$ (continuous line)
and $C_{L}(Q,\omega)$ (dashed lines) obtained in Ref. 66 from the IXS
measurement. The presence of a single mode exceeding the adiabatic frequency
(marked by the arrow) is clearly visible at $Q=2$ nm-1, i.e. well below the
crossover to the partial type dynamics. For $Q=6.5$ nm-1 two excitations
appears, at the high frequency one is dominating at $Q=9$ nm-1.

The presence of the two phonon-like modes in the IXS spectra at intermediate
$k$ can be conveniently quantified by looking at their relative weights
reported in Ref. 66 and presented in Fig. 27. The low frequency mode is
clearly dominant below the sharp crossover occurring at $Q\simeq 6$ nm-1.
Around this value, both modes contribute to the IXS spectra, while above it
the high frequency mode dominates up to a new crossover at $Q\simeq 13$ nm-1.
Since the IXS cross section is roughly proportional to the total density
autocorrelation function, below the crossover it reflects the collective
longitudinal excitation and not the optic-like mode. At small $Q$, the
relative weight of the optic-like propagating modes at small wavenumbers is
proportional to $Q^{2}$ as for any other collective excitation of the kinetic
origin.

Similar results have been obtained in the recent paper [65], where the high-
resolution IXS measurements were carried out on the molten NaI near the
melting point at 6800C. The measured spectra agreed well (in both frequency
and linewidth) with the ab initio molecular dynamics simulations but not with
the classical ones. The observation of these modes at small $k$ and a good
agreement (see Fig. 28) with the simulations permits their clear
identification as the collective optic-like excitations with the well defined
phasing between different ionic motions. Besides, the obtained data for the
dispersion relations and spectra linewidth fit perfectly into the theoretical
curves obtained within the GCM approach [69].

Figure 28: Dispersions of the collective propagating excitations in molten
NaI. Circles and triangles indicate the experimental data for the longitudinal
optic (LO) and acoustic (LA) modes, respectively, and the solid and dashed
curves show the theoretical ones [69].

Though the dynamic properties of ionic liquids and neutral mixtures are
similar and can be studied within the same theoretical and experimental
approaches, there are some peculiarities dealt with a non-zero contribution of
the optic-like excitations to the “charge-charge” dynamic structure factor
even in the hydrodynamic limit. Fortunately, in the case of the mixtures with
the long-range Coulomb interaction between their particles, the problem of
observation of the optic-like excitations can be solved even experimentally. A
theoretical description of these excitations in ionic liquids was a subject of
numerous studies, and up to now some controversies in the obtained results are
not explained.

####  5 Some rigorous relations and their applications

Let us return now to the expressions for the elements of memory functions
(21)-(25) that look rather sophisticated for practical applications. One of
the possibilities to proceed is to use them for calculation of the generalized
transport coefficients (see, for instance, Ref. 40). But there is also another
option that allows us to derive some rigorous relations being useful in
practice.

One of the most interesting results, important in the context of our
consideration of molten salts, was obtained in the middle of the XIX century
by Johann Wilhelm Hittorf. He reasonably has pointed out that some ions travel
more rapidly than others, and this observation led to the concept of the
transport number, the fraction of the electric current carried by each ionic
species. He measured the changes in the concentration of electrolyzed
solutions, computed from these transport numbers (relative carrying
capacities) of the ions, and established his laws governing the migration of
ions. A century later, B. R. Sundheim demonstrated [70] that simple
considerations of the conservation of electrolyte particles momentum lead to
definite predictions about the relative motions of parts of such ionic
complexes as measured in their particular rearrangement. It has been shown
that the transference numbers $t_{\alpha}$ of the species in a pure molten
salt can be expressed in terms of the masses $M_{\alpha}$ as
$t_{\alpha}=M_{\alpha}/\sum_{\nu}M_{\nu}$. Since every considered electrolysis
cell has to obey the condition of electroneutrality $t_{+}q_{+}-t_{-}q_{-}=0$,
and the electric current $J_{\alpha}$ is dealt with the electric field
strength $E$ by the Ohm’s law, $J_{\alpha}=\sigma_{\alpha}E$, one can
anticipate the simple relation between the partial electric conductivities
$\sigma_{\alpha}$,

$\frac{\sigma_{+}}{\sigma_{-}}=\frac{M_{-}}{M_{+}}.$ (57)

It was obvious that a simple phenomenological treatment of the ionic transport
in electrolytes did not allow to obtain anything more accurate than (57). To
move further the statistical mechanical theory had to be applied. In Ref. 71
such a study with aim to derive rigorously the relation (57) was performed for
the case of molten NaCl and NaI using the Langevin equations for motion of
cations and anions. The partial ionic conductivities were derived in terms of
the “current-current” TCF as an extension of the Kubo theory. It was shown
that the relation (57) is fulfilled in this approach, and for the total
conductivity $\sigma$ the following expression was found:

$\sigma=\sigma_{+}+\sigma_{-}=\frac{e^{2}}{k_{B}T}\left(n_{+}z_{+}^{2}D_{+}+n_{-}z_{-}^{2}D_{-}\right)(1-\Delta),$
(58)

where $e$ means the electron charge, $n_{\alpha}$ and $z_{\alpha}$ denote the
$\alpha$-ion ($\alpha=+,-$) number density and valence, respectively, and
$D_{\alpha}$ denotes the corresponding self-diffusion constant. The quantity
$\Delta$ is treated as the deviation from the Nernst-Einstein relation, being
defined by some cation-anion cross-correlation terms, involving the zeroth
time moments of the velocity intercorrelation functions. This $\Delta$
correction was evaluated in the MD simulations, and its absolute value turned
out to be smaller then unity in both cases of molten NaCl at $T=1100$ K and
NaI at $T=950$. However, for NaCl its value was strongly underestimated as
compared with the experimental result for $\Delta$, and for NaI it was twice
overestimated, whereas the theoretical results for the diffusion constants
were consistent with the experimental ones. In Ref. 71 the MD frequency
dependent partial conductivities were also calculated and their maxima were
found to occur almost at the same frequencies both for NaCl and NaI (see Fig.
29)

Figure 29: The frequency dependences of electrical conductivities in molten
NaI calculated in Ref. 71 by using MD (lines with symbols) and within the
theory using $\omega$-dependent self-diffusion coefficients (lines).

As follows from the comparison of the corresponding curves in Fig. 29, the
proposed theory does not allow to capture the main features of the frequency
dependence of either complete or partial conductivities. From the physical
point of view, this is understandable, because the frequency dependences of
the self-diffusion coefficients are largely determined by the Einstein
frequencies [49], which correspond to the average frequency of oscillations of
the tagged particle in the environment of others. It is clear that the
frequency of such oscillations will be different for each particle type,
because their masses are different, and so are their immediate environments,
which indirectly affect the oscillatory motion of the selected particle. Since
the maxima of the partial conductivities are localized at the same frequency,
it is obvious that this frequency characterizes the oscillatory motion
inherent to the system as a whole and cannot be described in one-particle
circuits. So what kind of collective motion is this? We will return to this
question a bit later.

In Ref. 72 the structural properties of the $\beta$ phase and transport
properties of the superionic $\alpha$, $\beta$, and molten phases777These
phases differ from each other because of the different arrangement of anions,
which can be observed in the corresponding pair distribution functions. of the
Ag3SI system were investigated by the MD simulations, using the
Vashishta–Rahman-type potentials. The most important conclusion of this study,
being interesting for our consideration, is the new relation proposed for the
partial conductivities

$\displaystyle
m_{\tiny\mbox{S}}\sigma_{\tiny\mbox{S}}/|z_{\tiny\mbox{S}}|+m_{\tiny\mbox{I}}\sigma_{\tiny\mbox{I}}/|z_{\tiny\mbox{I}}|=m_{\tiny\mbox{Ag}}\sigma_{\tiny\mbox{Ag}}/|z_{\tiny\mbox{Ag}}|.$
(59)

This relation can be considered as a generalized version of (57) for a ternary
mixture. Further studies of the pseudo-binary molten salts, performed by
Matsunaga and co-authors in Ref. 73 (AgI-AgBr system) and in Ref. 74 (NaCl-KCl
system), have shown that the relation (59) is valid for these ternary alloys.
It then became even more important to prove such “a golden rule” expression
within the non-equilibrium statistical mechanics approach.

In Sec. 3 the equations of the generalized hydrodynamics were derived using
the NSO method. It allowed us to obtain the expressions for generalized
transport coefficients (21)-(25). One of the important achievements of the
theory is a possibility to prove a number of exact relations for the
generalized transport coefficients immediately from the conservation laws. It
follows already from the structure of the expressions for the memory
functions, defined for the set of hydrodynamic variables and constructed on
the corresponding generalized currents. For the generalized currents $I_{{\bf
k},\alpha}^{d}$ related to the partial densities of particles
$\\{\hat{n}_{{\bf k},\alpha}\\}$, we have the identity

$\sum\limits_{\alpha=1}^{\nu}m_{\alpha}I_{{\bf
k},\alpha}^{d}=\sum\limits_{\alpha=1}^{\nu}\left(\hat{J}_{{\bf
k},\alpha}^{||}-\frac{m_{\alpha}c_{\alpha}}{\bar{m}}\hat{J}_{\bf
k}^{||}\right)\equiv 0,$ (60)

from which we directly obtain a number of exact relations for the generalized
transport coefficients of a $\nu$-component mixture:

$\sum\limits_{\alpha=1}^{\nu}m_{\alpha}D_{\alpha\gamma}(k,z)=\sum\limits_{\gamma=1}^{\nu}m_{\alpha}D_{\alpha}^{T}(k,z)=\sum\limits_{\alpha=1}^{\nu}m_{\alpha}\zeta_{\alpha}(k,z)\equiv
0.$ (61)

Among them, the mutual diffusion coefficients $D_{\alpha\gamma}$ are the most
important for our subsequent analysis. In the case of the $\nu$-component
fluid, the matrix of the mutual diffusion coefficients contains $\nu\times\nu$
elements. Taking into account the symmetry
$D_{\alpha\gamma}=D_{\gamma\alpha}$, their number is reduced to
$\nu\times(\nu+1)/2$ independent elements. And if we also take into account
the exact relations given above, we come to the conclusion that only
$\nu\times(\nu-1)/2$ independent elements remain. Thus, in a binary mixture
there is only one independent coefficient, in a ternary fluid its number rises
to three, etc.

Let us now consider an ionic system with the charges $q_{\alpha}$, masses
$m_{\alpha}$, and concentrations $c_{\alpha}$. In such a system the
electroneutrality condition $\sum_{\alpha}q_{\alpha}c_{\alpha}=0$ holds. For
the properly normalized mutual diffusion coefficient
$D_{\alpha\gamma}=c_{\alpha}c_{\beta}\bar{D}_{\alpha\gamma}$, the exact
relation for $\bar{D}_{\alpha\gamma}$ can be rewritten as

$\sum\limits_{\alpha=1}^{\nu}m_{\alpha}c_{\alpha}\bar{D}_{\alpha\gamma}(k,\omega)=\sum\limits_{\gamma=1}^{\nu}\bar{D}_{\alpha\gamma}(k,\omega)c_{\gamma}m_{\gamma}\equiv
0.$ (62)

If an external field ${\bf E}_{\omega}$ is applied to the system, an
expression for the ionic conductivity can be derived within the linear
response theory. For the partial conductivities one gets:

$\sigma_{\alpha}(\omega)=k_{0}q_{\alpha}c_{\alpha}\sum\limits_{\beta}q_{\beta}c_{\beta}\bar{D}_{\alpha\beta}(\omega),$
(63)

where $k_{0}$ is a certain coefficient depending only on the temperature and
density of the system. If we multiply $\sigma_{\alpha}(x,\omega)$ by
$m_{\alpha}/q_{\alpha}$ and perform the summation over $\alpha$, the general
relation for partial conductivities can be obtained:

$\displaystyle\sum\limits_{\alpha}\frac{m_{\alpha}}{q_{\alpha}}\
\sigma_{\alpha}(\omega)=k_{0}\sum\limits_{\alpha}m_{\alpha}c_{\alpha}\left[\sum\limits_{\beta}q_{\beta}c_{\beta}\bar{D}_{\alpha\beta}(\omega)\right]$
$\displaystyle=k_{0}\sum\limits_{\beta}q_{\beta}c_{\beta}\left[\sum\limits_{\alpha}m_{\alpha}c_{\alpha}\bar{D}_{\alpha\beta}(\omega)\right]\equiv
0,$ (64)

and the identical equality to zero in (5) follows from Eq. (62).

A more complicated case corresponds to the model of the
($\nu+\bar{\nu}$)-component fluid that is composed of $N_{\alpha}$ ions with
charges $q_{\alpha}$ ($\alpha=1,2,\ldots,\nu$) and $N_{\bar{\alpha}}$ neutral
particles belonging to the $\bar{\alpha}$-th species
($\bar{\alpha}=1,2,\ldots,\bar{\nu}$). In this case, Eq. (5) should be
rewritten in the form

$\displaystyle\sum\limits_{\alpha=1}^{\nu}\frac{m_{\alpha}}{q_{\alpha}}\
\sigma_{\alpha}(\omega)=k_{0}\sum\limits_{\beta=1}^{\nu}q_{\beta}c_{\beta}\left[\sum\limits_{\alpha=1}^{\nu}m_{\alpha}c_{\alpha}\bar{D}_{\alpha\beta}(\omega)\right]$
$\displaystyle=-k_{0}\sum\limits_{\beta=1}^{\nu}q_{\beta}c_{\beta}\left[\sum\limits_{\bar{\alpha}=1}^{\bar{\nu}}m_{\bar{\alpha}}c_{\bar{\alpha}}\bar{D}_{\bar{\alpha}\beta}(\omega)\right],$
(65)

where the relations (62) have been used.

The expression (5) can be further simplified if we introduce two new
densities, namely the mass density of solvent $\hat{M}_{\bf k}$ (formed by
neutral particles only) and the charge density $\hat{Q}_{\bf k}$:

$\hat{M}_{\bf
k}=\sum\limits_{\bar{\alpha}=1}^{\bar{\nu}}m_{\bar{\alpha}}c_{\bar{\alpha}}\hat{n}_{\bar{\alpha},{\bf
k}}\,,\qquad\hat{Q}_{\bf
k}=\sum\limits_{\alpha=1}^{\nu}q_{\alpha}c_{\alpha}\hat{n}_{\alpha,\bf k}\,.$
(66)

Now we can rewrite (5) in the form

$\displaystyle\sum\limits_{\alpha=1}^{\nu}\frac{m_{\alpha}}{q_{\alpha}}\
\sigma_{\alpha}(\omega)=-D_{\rm MQ}(\omega),$ (67)

where the generalized transport coefficient

$D_{\rm
MQ}(\omega)=k_{0}\sum\limits_{\bar{\alpha}=1}^{\bar{\nu}}\sum\limits_{\beta=1}^{\nu}\
m_{\bar{\alpha}}c_{\bar{\alpha}}\ \bar{D}_{\bar{\alpha}\beta}(\omega)\
q_{\beta}c_{\beta}$ (68)

describes the diffusive ion-solvent cross-correlations. The relation (67) is,
in fact, the most general form of the “golden rule” identity valid for partial
ionic conductivities of classical systems of charged particles in a neutral
solvent. As it has been shown this relation follows directly from the
conservation law for the total momentum (see (62)) and the electroneutrality
condition. Obviously, the expressions (57) and (59) can be easily obtained
from (67).

Returning to the question about the frequency dependences of partial and total
conductivities (see Fig. 29), we note that
$\sigma_{1}(\omega)/\sigma_{2}(\omega)=|q_{1}|m_{2}/(|q_{2}|m_{1})$ and this
allows: i) to explain why the peak for all the conductivities obtained in the
MD simulations is localized at the same frequency; ii) to determine the origin
of this excitation. The last statement follows from the fact that frequency-
dependent conductivities in a molten salt can be expressed via the spectral
functions of the charge fluxes in the hydrodynamic limit. As it has been shown
earlier, at small $k$ the main contribution to these functions gives a
collective mode of optical type. The cases when the system in a new phase
state changes its conducting properties may be of a particular interest.

####  6 Dynamic crossover in supercritical fluids

One of the most attractive fields for the soft matter theory is the dynamics
of supercritical fluids. It is well known that upon increasing the substance
pressure and temperature beyond its critical point, any thermodynamic
discontinuity between the liquid and gas phase disappears and the system is
said to be in a fluid state. A character of the local ordering as well as the
physical properties of a system are changing significantly herewith. There are
some phenomena related to the collective dynamics of fluids in the
supercritical region that completely differ from those known for the systems
at more moderate conditions. For instance, the dispersion curve of the
acoustic excitation in dense liquids is found to deviate from the linear
relation $c_{s}k$ at small wavenumbers either upwards (yielding the so-called
“positive dispersion” [76]) or downwards (“negative dispersion” [77]).

To date two main mechanisms are recognized to be responsible for the positive
dispersion in fluids: nonlocal coupling between hydrodynamic modes, described
within the mode-coupling theory (MCT) [78], and local coupling between
acoustic excitations and non-hydrodynamic structural relaxation obtained
within the memory function formalism [79]. However, there was no systematic
theoretical and experimental study of the density or pressure dependence of
positive or negative deviation from hydrodynamic dispersion law, except for
the inelastic neutron scattering experiments and subsequent calculations,
based on MCT for liquid Ar at 120 K. The MCT yields the following expression
for the dispersion law of sound excitations

$\omega(k)=c_{s}k+\alpha_{s}k^{5/2}+O(k^{11/4})+\ldots,$ (69)

where the coefficient $\alpha_{s}$ is expressed in a sophisticated way,
requiring the knowledge of the explicit dependences of adiabatic sound
velocity $c_{s}$ and thermal expansion coefficient $\alpha_{T}$ on the
density. The results of Ref. 80 have proven the positivity of the prefactor
$\alpha_{s}$; however, little is known about its density dependence. The
recent IXS experiments on supercritical Ar [69] resulted in a completely
different tendency for the pressure dependence of the positive sound
dispersion in the supercritical region. It was observed that the positive
dispersion reduces with decreasing density and finally practically vanishes.

In this subsection, we present the results of Ref. 81, where the thorough
analysis of the sound dispersion along several isothermal lines on the phase
diagram of the supercritical Lennard-Jones fluids was performed within the GCM
approach. The analytical expressions, based on the local-coupling mechanism,
were obtained and used for the subsequent investigation of microscopic
processes, being responsible for the positive (or negative) sound dispersion
at the boundary of the hydrodynamic regime.

Figure 30:  Dispersion of collective sound excitations for different densities
at $T^{*}=4.78$ (left) and $T^{*}=1.71$ (right). Dashed line shows the linear
hydrodynamic dispersion law with the hydrodynamic sound velocity $c_{s}$[81].

In Fig. 30, the dispersion curves, calculated in Ref. 81 for several
densities, are shown at the reduced temperatures $T^{*}=4.78$ and 1.71. The
general tendencies in the dispersion change with density at $T^{*}=4.78$ are:
i) the reduction of the slope of the linear dispersion law at small $k$ with
decreasing density; ii) the reduction of the roton-like minimum with
decreasing density; iii) the reduction of positive dispersion with decreasing
density; and iv) increasing of the width of region, where the apparent sound
velocity almost coincides with $c_{s}$, with decreasing density.

It is interesting that at some low-density states the eigenvalues with a
negative dispersion of collective sound excitations are identified in our
study. These cases can be found at the right panel of Fig. 30 in the long-
wavelength region.

Using the thermo-viscoelastic dynamic model within the GCM approach the
analytical result for the sound dispersion in the small $k$ domain has been
obtained:

$\omega(k)=c_{s}k+\beta_{s}k^{3}+\ldots,$ (70)

where the coefficient

$\beta_{s}=\frac{c_{s}D_{L}^{2}}{8}\,\frac{5-(c_{\infty}/c_{s})^{2}}{c_{\infty}^{2}-c_{s}^{2}}-(\gamma-1)D_{T}\left[\frac{6D_{L}+(\gamma-5)D_{T}}{8c_{s}}-\frac{c_{s}}{2d_{T}}\right]$
(71)

is expressed via the adiabatic $c_{s}$ and high-frequency $c_{\infty}$ sound
velocities, the ratio $\gamma$ of specific heats at constant pressure and
constant volume, the kinematic viscosity $D_{L}$, the thermal diffusivity
$D_{T}$, and the long-wavelength limit $k\to 0$ of the kinetic structural
relaxation mode $d_{T}$. Obviously, the expression (70) gives a different
result from (69), found by the MCT. Note that the first term in (70) describes
the viscoelastic contribution, whereas the second term reflects mainly the
thermal processes and vanishes if $\gamma$ tends to zero.

Figure 31:  Dependence of factor $\beta_{s}$, Eq. (71) on the density for two
temperatures $T^{*}=4.78$ and $T^{*}=1.71$ (closed boxes). Open circles
correspond to viscoelastic contribution to the $\beta_{s}$[81].

In Fig. 31, the results of numerical calculations of the factor $\beta_{s}$
from Eq. (71) at two temperatures using the GCM data as an input are
presented. The strength of the contribution from the first term in (71) is
shown by open circles. The evident difference between the total and
viscoelastic contributions is attributed to the thermal processes (the second
term in Eq. (71)). One can see that for the high-temperature state
$T^{*}=4.78$ the positive dispersion is completely defined by the viscoelastic
mechanism. Most interesting is the fact that the positive dispersion
practically vanishes for that temperature around $n^{*}\approx 0.8$. For lower
(and closer to the critical point) temperature $T^{*}=1.71$, the effect of the
thermal contribution is much stronger. At low temperatures for densities
$n^{*}>0.75$ the dispersion curve in the long-wavelength region shows a
positive dispersion, while for $n^{*}<0.6$ the dispersion can be negative in a
narrow region of wavenumbers close to the hydrodynamic regime. By comparing
the results obtained for the factor $\beta_{s}$ (see Fig. 31) that defines the
deviation from linear dispersion law in the long-wavelength region with the
dispersion curves, calculated within the GCM approach and shown in Fig. 30,
one can conclude that the analytic expression (70) describes the sound
dispersion at both densities. Therefore, there is an additional argument to
state that the main mechanism responsible for deviation of the sound
dispersion from the linear dependence at small $k$ is dealt with the local
interaction of collective modes, which is easy to take into account within the
GCM approach. Moreover, the beginning of the range with the subsequent growth
of the viscoelastic term in $\beta_{s}$ (see Fig. 31) can be used as a simple
criterion for estimation of the crossover from the gas-like fluid to the
liquid-like fluid. A role of the non-hydrodynamic processes in the
viscoelastic transition in pure fluids with the main emphasis on the relation
between the appearance of shear waves in the transverse dynamics and the
manifestation of positive sound dispersion in the longitudinal dynamics was
studied recently in Ref. 55, but this might be a topic for another review.

###  6 Conclusions

In this Chapter we have considered some old and new problems in the fluid
dynamics, which are interesting from both theoretical and experimental points
of view. All of them are united by the same subject of research and the
theoretical approach applied. The fluids described in this chapter differ from
each other by their constituents (single-component, binary, and ternary
systems), types of interactions between the particles (Lennard-Jones, Coulomb
and many others), and thermodynamic states considered. At the same time we
tried to demonstrate that at least some of these problems can be efficiently
attacked within the GCM approach, because: i) it is based on the rigorous
theoretical framework that creates the NSO method; ii) it allows to treat
effectively the time/space hierarchy of the processes by choosing an
appropriate set of dynamical variables; iii) it is non-perturbative and does
not use any fitting parameters (the only input data are the SCFs and the
hydrodynamic relaxation times, which can be taken from computer simulations).
As the output, one obtains the spectrum of single-particle or collective
excitations, which enter the corresponding TCFs additively, and each mode
contribution can be traced accurately. At the same time, the results for TCFs
obtained within the GCM approach are controlled by the sum rules up to some
fixed order and found to be close to those obtained experimentally, in
particular in the scattering experiments.

The extended set of dynamical variables complements the hydrodynamic
collective mode spectrum due to the appearance of kinetic modes, which have
finite damping coefficients in the hydrodynamic limit. The criteria for these
modes to be either purely relaxational or of a propagative nature are
formulated. In the latter case, the new phenomena like shear waves or fast
sound have been predicted and observed both in the experiment and in computer
simulations. Sometimes the contributions from the obtained propagative kinetic
modes are overlapped by the terms from the generalized hydrodynamic modes. In
other circumstances they are well separated from the hydrodynamics mode
spectrum and at modest values of the damping can be quite distinguishable at
the corresponding TCFs. We have also pointed out the conditions when the fluid
dynamics has a “collective” nature and can be best modelled by the mass
density and mass concentration variables, or it is of “partial” nature, being
dominantly described by the densities of particular species of the fluid. The
crossover from the collective behaviour to the partial one has a rather
general character.

Several simple (but non-trivial) dynamical models have been considered and
analysed in detail. It has been shown that these models can be solved
analytically, and the obtained results are very useful for establishing the
nature of the generalized collective modes and deriving criteria for their
appearance and/or experimental observations. This allows us to study how
fluids behave at various time scales and describe the crossover from the
dynamics typical for a Newtonian fluid to the elastic solid-like behaviour. It
has been shown that even at the level of the single-particle dynamics the
collective cage effects can play an important role. For binary fluids the
criterium for the appearance of the optic-like propagating modes has been
formulated and tested in computer simulations.

We have also presented some rigorous relations for the transport coefficients
of a multicomponent fluid. These relations in the case of a system of charged
particles allowed us to derive rigorously the general version of the so-called
“golden rule” for partial conductivities. It is shown that such a relation can
be used as a new opportunity to observe the optic-like modes in a fluid
mixture.

Having read the above list of possibilities and merits of the GSM approach,
one can ask some natural questions: is this method suitable for studying the
collective dynamics of fluid in the entire space and time domains? What is the
relation of the GCM method to other theoretical approaches? Is there any
limitation of its applicability for the description of fluid dynamics?

Surely, the collective dynamics of fluids can be best described by those
approaches that rest upon strict statistical mechanics relations, which allow
generalization of both the thermodynamics and hydrodynamics, and are amenable
to computer simulations. In this context, the GCM approach should be
considered as one of the most developed options that can be used in
combination with others depending on the problem considered. Still, there
remain plenty of questions about its applicability to the dynamics of complex
fluids such as solutions of macromolecules and proteins. Hence, answering old
question and solving some old puzzles lead to new ones, to be solved soon.

## References

  * 1. M. Krasnytska, R. de Regt, P. Sarkanych, eds., _Ising Lectures in Lviv (1997–2017)_. ICMP of National Academy of Sciences of Ukraine, Lviv (2017).
  * 2. A. G. Bedeian, A note on the aphorism “there is nothing as practical as a good theory”, _J. Manag. History_. 22(2), 236–242 (2016).
  * 3. I. M. Mryglod, I. P. Omelyan, M. V. Tokarchuk, Generalized collective modes for the Lennard-Jones fluid, _Mol. Phys._ 84(2), 235–259 (1995).
  * 4. I. M. Mryglod, Generalized hydrodynamics of multicomponent fluids, _Condens. Matter Phys._ 10, 115–135 (1997).
  * 5. I. M. Mryglod, Generalized statistical hydrodynamics of fluids: Approach of generalized collective modes, _Condens. Matter Phys._ 1(4), 115–135 (1998).
  * 6. D. N. Zubarev, _Nonequilibrium statical thermodynamics_. Consultant Bureau, New–York (1974).
  * 7. D. Zubarev, V. Morozov, G. Röpke, _Statistical mechanics of nonequilibrium processes: 1. Basic covncepts, kinetic theory_. Academie Verlag, Berlin (1997).
  * 8. D. Zubarev, V. Morozov, G. Röpke _Statistical mechanics of nonequilibrium processes: 2. Relaxation and hydrodynamic processes_. Academie Verlag, Berlin (1997).
  * 9. A. N. Beris, A. J. Giacomin, $\Pi\alpha\nu\tau\alpha$ $\rho\epsilon\iota$: Everything Flows, _J. Appl. Rheol._ 24, 52918:1-13 (2014).
  * 10. D. Christodoulou, The Euler equations of compressible fluid flow, _Bull. Amer. Math. Soc._ 44(4), 581–602 (2017).
  * 11. S. R. de Groot, P. Mazur, _Non-equilibrium thermodynamics_. North-Holland, Amsterdam, Dover edition (1984).